UC-NRLF 


SB    E7fl 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


GIFT    OF 


3.r.jX.r...^ 

Class 


\^M\^(^^^^ 


LESSONS 


IN 


GEOMETKY 


FOR  THE  USE  OF  BEGINNERS. 


BY 

G.  A.  HILL,  A.M., 

AUTHOR  OF  A  GEOMETRY  FOR  BEGINNERS. 


BOSTON: 

PUBLISHED  BY   GINN   &   COMPANY. 

1888. 


Entered,  according  to  Act  of  Congress,  in  the  year  1887,  by 

G.  A.  HILL, 
in  the  office  of  the  Librarian  of  Congress,  at  Washington 


ELECTROTYPED  BY  J.  8.  CUBBING  &  Co.,  BOSTON. 


BY   GlNN  &  Co.,   BOSTON. 


PREFACE. 


THIS  book  has  been  prepared  to  meet  the  wishes  of  those  who 
prefer  a  shorter  and  easier  introductory  course  in  Geometry  than 
that  given  in  the  "  Geometry  for  Beginners." 

The  metric  system  of  units  is  explained;  but  all  exercises  in 
metric  units  are  confined  to  lessons  placed  at  the  ends  of  the 
chapters,  and  may  be  omitted  if  desired. 

The  method  of  instruction  is  that  best  adapted  to  the  mental 
condition  of  pupils  between  the  ages  of  twelve  and  sixteen.  The 
training  in  consecutive  reasoning  is  introduced  very  gradually, 
and  is  confined  mainly  to  the  laws  of  equal  triangles  and  a  few 
of  their  simple  applications. 

To  one  feature  of  the  method  the   author  desires  to  call  spe- 
cial   attention ;   namely,  the    numerous   exercises    which    involve 
the   use   of  instruments   and  drawing  to   scale.     It   is   assumed 
that    every  pupil  is   provided  with   ruler,   divided   scale,   pencil 
compasses,  triangle,  and  protractor.     Any  objection  on  the  ground  f 
of  expense  has  been  met  by  the  publishers,  who  are  prepared  to 
supply,  at  a  very  low  price,  these  instruments  enclosed  in  a  strong  j 
wooden  box. 

Experience  shows  that  for  the  beginner  of  Geometry  the  care- 
ful execution  of  easy  constructions  is  the  most  useful  as  well  as 
the  most  interesting  part  of  the  daily  lesson.  This  work  calls 
into  action  the  eye,  the  hand,  and  the  judgment.  It  holds  the 
attention.  It  is  attended  with  the  pleasing  sense  of  the  successful 
exercise  of  new-found  power.  Under  these  conditions  progress  in 
knowledge  is  sure  to  be  rapid.  The  precept,  "  Do  that  you  may 
know,"  finds  here  a  pertinent  application. 

183667 


IV  PREFACE. 

No  teacher,  however,  can  expect  to  obtain  the  best  results  from 
this  kind  of  work,  unless  he  insists  strenuously  upon  neatness  and 
a  reasonable  degree  of  accuracy.  Every  teacher  should  lay  down 
a  standard  of  neatness  and  accuracy,  —  remembering  that  his 
pupils  are  only  beginners  working  with  cheap  instruments,  —  and 
then  should  criticise  unsparingly  every  drawing  which,  tried  by 
his  standard,  is  a  slovenly  or  inaccurate  piece  of  work. 

The  contents  of  the  book  are  so  arranged  that  the  course  may 
be  considerably  abridged,  if  so  desired.  There  are,  in  all,  ninety- 
six  lessons  and  fifteen  drawing  exercises.  This  makes  abundant 
material  for  a  course  of  three  hours  per  week  for  a  year,  or, 
what  is  better  for  the  pupil,  a  course  of  one  hour  per  week  for 
a  year,  and  a  course  of  two  hours  per  week  for  the  year  follow- 
ing. The  last  two  chapters,  however,  may  be  omitted,  and  like- 
wise the  lessons  in  which  metric  units  are  used ;  there  will 
then  be  left  sixty-five  lessons  and  the  drawing  exercises,  or  a 
course  of  two  hours  per  week  for  one  year. 

Geometry,  as  here  presented,  should  be  studied  before  Algebra. 
If  this  is  done,  pupils,  while  learning  the  properties  of  figures  and 
the  measurement  of  areas  and  volumes,  will  see  for  themselves 
the  great  advantage  of  using  letters  to  represent  quantities. 
Thus  the  chief  stumbling-block  to  every  beginner  of  Algebra  will 
be  removed. 

The  author  takes  this  opportunity  to  express  his  warm  thanks 
to  the  teachers  who  have  had  the  kindness  to  read  and  correct 
the  proof-sheets.  A  special  acknowledgment  of  obligation  for 
this  assistance  is  due  to  Prof.  G.  A.  WENTWORTH,  of  Exeter, 
N.H.;  Prof.  H.  D.  WOOD,  of  Trenton,  Ga. ;  Mr.  J.  E.  CLARKE, 
of  Chelsea,  Mass. ;  and  Mr.  E.  H.  NICHOLS,  of  Cambridge,  Mass. 

G.  A.  HILL. 

CAMBRIDGE,  Feb.  29,  1888. 


CONTENTS. 


CHAPTER   I.  — INTRODUCTION. 
LESSON  PAGE 

1.  Observation  of  a  cube  and  a  cylinder 1 

2.  Questions  about  prisms,  pyramids,  etc 4 

3.  Body,  surface,  line,  point 5 

4.  Different  kinds  of  lines,  surfaces,  and  bodies 8 

CHAPTER   II.  — STRAIGHT  LINES. 

5.  Determination  of  straight  lines.    Use  of  the  ruler 10 

6.  Vertical,  horizontal,  and  inclined  lines 12 

7.  The  circle.     Use  of  the  compasses 15 

8.  Parallel  lines,  and  their  construction 18 

9.  Equal  and  unequal  lines.     Axioms  of  Geometry 21 

10.  Addition,  subtraction,  etc.,  of  lines .24 

11.  Units  of  lengths 27 

12.  Measurement  of  straight  lines 29 

18.  Drawing  to  scale 30 

14.  The  metric  units  of  length 32 

15.  Drawing  and  measuring  with  metric  units 34 

16.  Review  of  Chapter  II.  begun 35 

17.  Review  of  Chapter  II.  concluded 36 

CHAPTER   III. —ANGLES. 

18.  Definitions.     The  magnitude  of  an  angle 37 

19.  Erecting  perpendiculars 40 

20.  Dropping  perpendiculars 42 

21.  Measurement  of  angles.     Use  of  the  protractor 43 

22.  Construction  of  angles.     Bisecting  an  angle 46 

23.  Adjacent  angles  and  vertical  angles 48 

24.  Two  parallel  lines  cut  by  a  third  line 50 


VI  CONTENTS. 

LESSON  PAGE 

25.  Review  of  Chapter  III.  begun 52 

20.  Review  of  Chapter  III.  continued 53 

27.  Review  of  Chapter  III.  concluded  (metric  units)  .....  54 

CHAPTER   IV.— TRIANGLES. 

28.  Definitions.     Dimensions  of  a  triangle 55 

29.  Construction  of  triangles.     Practical  applications      ....  58 

30.  Construction  of  triangles.     Practical  applications      ....  60 

31.  Sum  of  the  angles  of  a  triangle 62 

32.  Equivalence,  similarity,  and  equality 64 

33.  Theorems.     Method  of  equal  triangles 67 

34.  Theorems 70 

35.  Theorems 72 

36.  Review  of  Chapter  IV.  begun 74 

37.  Review  of  Chapter  IV.  continued 75 

38.  Review  of  Chapter  IV.  continued 7(5 

39.  Review  of  Chapter  IV.  continued  (metric  units) 77 

40.  Review  of  Chapter  IV.  concluded  (metric  units) 78 

CHAPTER   V.  — POLYGONS. 

41.  Definitions.     Sum  of  the  angles  of  a  polygon 79 

42.  Dimensions  of  quadrilaterals.     Constructions 82 

43.  Parallelograms.     Theorems  and  constructions 84 

44.  Parallelograms.     Theorems  and  constructions 86 

45.  Polygons.   Miscellaneous  constructions 88 

46.  Regular  polygons 90 

47.  Review  of  Chapter  V.  begun 92 

48.  Review  of  Chapter  V.  continued 93 

49.  Review  of  Chapter  V.  concluded  (metric  units) 94 


CHAPTER   VI.  — THE   CIRCLE. 

50.  Definitions.     Chords,  arcs,  and  angles  at  the  centre  ....  95 

51.  Radius  perpendicular  to  a  chord.     Constructions 98 

52.  Inscribed  angles 100 

53.  Tangents 102 

54.  Two  circles.     Circumscribed  and  inscribed  figures     ....  104 


CONTENTS.  Vii 

LESSON  PAGE 

55.  Centre  of  a  regular  polygon.     Constructions 106 

56.  Circumference  of  a  circle.     Meaning  of  w 108 

57.  Review  of  Chapter  VI.  begun 110 

58.  Review  of  Chapter  VI.  continued Ill 

59.  Review  of  Chapter  VI.  concluded  (metric  units) 112 

CHAPTER   VII.— AREAS. 

60.  Units  of  area.     Area  of  a  square 113 

61.  Area  of  a  rectangle 116 

62.  Area  of  a  parallelogram 118 

63.  Area  of  a  triangle 120 

64.  Area  of  a  polygon 122 

65.  Area  of  a  circle 124 

66.  Theorem  of  Pythagoras - 126 

67.  Mean  proportional.     Transformation  of  figures 128 

68.  Transformation  of  figures 130 

69.  Review  of  Chapter  VII.  begun 132 

70.  Review  of  Chapter  VII.  continued 133 

71.  Review  of  Chapter  VII.  continued 134 

72.  Review  of  Chapter  VII.  continued 135 

73.  Review  of  Chapter  VII.  continued  (metric  units)      ....  136 

74.  Review  of  Chapter  VII.  concluded  (metric  units)      ....  138 

CHAPTER  VIII.  —  RATIOS. 

75.  Definitions.     Triangles  and  parallelograms  compared    .     .     .  139 

76.  Pivision  of  figures  into  parts 142 

77.  Proportions.     Application  to  the  circle 144 

78.  Similar  triangles.     Ratio  of  similitude 146 

79.  Applications.     Areas  of  two  similar  triangles 148 

80.  Numerical  relations  in  right  triangles 150 

81.  Review  of  Chapter  VIII.  begun 152 

82.  Review  of  Chapter  VIII.  continued 153 

83.  Review  of  Chapter  VIII.  concluded  (metric  units)    ....  154 

CHAPTER   IX.  — SOLIDS. 

84.  Planes  and  dihedral  angles 155 

85.  The  cube 158 

80.   The  rectangular  solid 160 


Vlll 


CONTENTS, 


LESSON 


PAGE 


87.  The  right  prism 162 

88.  The  right  cylinder 164 

89.  The  right  pyramid .  100 

90.  The  right  cone .     .'  108 

91.  The  sphere 170 


The  sphere  concluded 172 

Review  of  Chapter  IX.  begun 174 


92. 
93. 

94.  Review  of  Chapter  IX.  continued 175 

95.  Review  of  Chapter  IX.  continued  (metric  units) 170 

90.   Review  of  Chapter  IX.  concluded  (metric  units) 178 

Drawing  Exercises 179 


SIGNS  AND  ABBREVIATIONS. 


+,   increased  by. 
— ,    diminished  by. 
X,   multiplied  by. 
-h,    divided  by. 
=,   is  (or  are)  equal  to. 
>,  is  (or  are)  greater  than. 
<,  is  (or  are)  less  than. 
II,   parallel. 
_L,  perpendicular. 
y' ,  the  square  root  of. 


Z,  angle. 

A,  angles. 

A,  triangle. 

A,  triangles. 
Ax.,  axiom. 
T)ef.,  definition. 
Hyp.,  hypothesis. 
Con.,  conclusion. 
Const.,  construction. 

rt.,  right. 


LESSONS   IN   GEOMETRY 


CHAPTER   I. 


INTRODUCTION. 


Lesson  1. 

1.  Why  is  this  block  of  wood  (Fig.  1)  called  a  body  ? 
Why  is  it  also  called  a  cube? 

The  block  is  called  a  body  because  it  occupies  space, 
or,  in  other  words,  "takes  up  room."  Everything  which 
occupies  a  portion  of  space  is  called  a  body. 

The  block  is  called  a  cube  on  account  of  its  shape.  Every 
bod\'  having  the  shape  of  this  block  is  called  a  cube. 

2.  Point  out  and  name  its  dimensions. 
The  block  can  be  measured  in  three 

main  directions  : 

(1)  From  left  to  right ; 

(2)  From  front  to  back  ; 

(3)  From  top  to  bottom. 

These  three  measurements  are  called 
the  dimensions  of  the  block  ;   one  is  FlG- 

called  the  length,  another  the  breadth,  the  other  the  thickness. 

The  dimension  which  is  measured  straight  upwards  or 
downwards  is  also  called  the  height  of  the  block. 


2  LESSONS   IN   GEOMETRY. 

3.  Are  the  dimensions  of  a  cube  equal  or  unequal  ? 

The  three  dimensions  of  this  cube  (Fig.  2)  seem  to  be 
equal.  If  we  measure  them  with  a  foot  rule,  we  shall  find 
that  they  are  equal.  The  dimensions  of  a  cube  are  equal. 

4.  Describe  the  surface  of  the  cube. 

The  cube  is  limited  or  bounded  by  its  surface.     When  we 
look  at  the  cube,  it  is  its  surface  only 
which  we  see.     When  we  handle  the 
cube,  it  is  its  surface  only  which  we 
touch.     The  surface  of  the  cube  con- 
sists of  six  distinct  parts,  called  faces: 
The  front  and  the  back  faces ; 
The  right  and  the  left  faces  ; 
FIG.  2.  The  upper  and  the  lower  faces. 

The  faces  have  no  thickness ;  each  face  has  only  two 
dimensions,  length  and  breadth,  or  length  and  height.  The 
faces  of  the  cube  are  flat  or  plane  surfaces. 

5.  Describe  the  edges  of  the  cube. 

Each  face  of  the  cube  is  bounded  by  four  edges. 
Each  edge  is  the  place  where  two  of  the  faces  meet. 
There  are  in  all  twelve  edges. 

The  edges  of  the  cube  are  lines;  they  have  neither  breadth 
nor  thickness  ;  they  have  only  one  dimension,  length. 
The  edges  of  the  cube  are  straight  lines. 

6.  Describe  the  corners  of  the  cube. 

• 
Each  edge  of  the  cube  is  limited  by  two  corners. 

Each  corner  is  the  place  of  meeting  of  three  edges. 
There  are  in  all  eight  corners. 

The  corners  of  the  cube  are  points;  they  have  no  dimen- 
sions, neither  length,  breadth,  nor  thickness. 
They  have  only  place  or  position. 


INTRODUCTION. 


3 


FIG.  3. 


7.  The   body  represented  in  Figure  3  is  called  a 
cylinder.    Point  out  and  name  its  dimensions. 

The  cylinder  has  three  dimensions.  The  di- 
mension measured  straight  upwards  or  down- 
wards is  called  the  length  or  height.  The  other 
dimensions  are  called  the  breadth  and  thickness  ; 
the}'  are  measured,  one  of  them  from  left  to 
right,  the  other  from  front  to  back. 

The   breadth   is  not   everywhere   the   same ; 
neither  is  the  thickness.     But  in  such  cases  we 
always   choose    the   greatest    breadth    and    the 
greatest  thickness.     The  greatest  breadth  and  the  greatest 
thickness  are  equal,  as  we  can  see  by  measuring  them  with  a 
foot  rule. 

8.  Describe  the  surface  of  the  cylinder. 

The  cylinder  is  bounded  on  all  sides  by  its  surface.  The 
surface  has  no  thickness  ;  it  has  only  two  dimensions,  length 
and  breadth. 

The  surface  consists  of  three  parts :  two  flat  or  plane  sur- 
faces, called  the  bases ;  and  a  surface  extending  all  around 
the  cylinder  between  the  bases,  called  the  convex  surface. 

The  convex  surface  is  not  flat ;  it  is  a  curved  surface. 

9.  Describe  the  edges  of  the  cylinder. 

The  cylinder  has  two  edges.  These  edges  are  the  bounda- 
ries of  the  bases.  They  are  also  the  places  where  the  lateral 
surface  meets  the  bases. 

The  edges  are  lines,  and  have  one  dimension,  length. 

These  edges  are  curved  lines. 

The  cylinder  has  no  corners. 

10.  Measure  with  a  foot  rule  the  dimensions  of  the  cylin- 
der wThich  is  given  you. 


4  LESSONS   IN   GEOMETRY. 

Lesson  2. 

1.  The  body  represented  in  Figure  4  is  called  a  prism. 
Point  out  and  name  its  dimensions.     Are  they  equal  or 

unequal?  Which  is  the  longest?  Which  the  shortest? 
Describe  carefully  the  surface,  the  edges,  and  the  corners,  as 
those  of  the  cube  were  described  in  Lesson  1. 

2.  Examine  in  like  manner  the  four-sided  prism  (Fig.  5). 

3.  Examine  in  like  manner  the  six-sided  prism  (Fig.  6). 

4.  Examine  in  like  manner  the  four-sided  pyramid  (Fig.  7) . 

5.  Examine  in  like  manner  the  cone  (Fig.  8). 

6.  Examine  in  like  manner  the  frustum  of  a  cone  (Fig.  9). 

7.  Examine  in  like  manner  the  sphere  (Fig.  10). 


FIG.  8. 


FIG.  9. 


INTRODUCTION.  5 

<J  ~  >    - 

Lesson  3. 

1,  What  is  a  bndy.  as  understood  in  Geometry  ? 

A  body,  in  common  language,  is  a  portion  of  space  filled 
with  something  which  we  can  see,  touch,  handle,  etc.  ;  for 
example  :  a  pencil,  a  book,  a  stone,  the  sun.  But  in  Geome- 
try we  pa}7  no  attention  to  the  matter  of  which  a  bod}'  is 
composed ;  we  study  simply  its  size  and  its  shape.  In 
Geometry,  a  body  means  simply  a  limited  portion  of  space. 

Geometrical  bodies  are  sometimes  called  solids, 

2,  How  many  dimensions  has  a  body?    How  are 
they  named? 

Every  body  has  three  dimensions.  The  words  used  for 
naming  dimensions  are  length,  breadth  (or  ividth) ,  thickness, 
height,  and  depth.  Usually  the  longest  dimension  is  called 
the  length,  and  the  shortest  the  thickness.  The  terms  height 
and  depth  are  used  only  for  dimensions  measured  straight 
upwards  or  downwards. 

3.  Define  the  terms  surface,  line,  and  point. 

A  surface  is  a  space  magnitude  having  only  two  dimen- 
sions. 

A  line  is  a  space  magnitude  having  only  one  dimension. 
A  point  is  a  position  in  space  without  magnitude. 
The  boundaries  of  a  body  are  surfaces. 
The  boundaries  of  a  surface  are  lines. 
The  limits  of  a  line  are  points. 

4.  Define  the  intersection  of  two  surfaces. 

When  two  surfaces  meet  or  cut  each  other,  the  place  where 
they  meet  is  called  their  intersection. 

The  intersection  of  two  surfaces  is  ahvays  a  line. 

For  example  :  the  faces  of  the  cube  intersect  in  the  edges. 


6  LESSONS    IN    GEOMETRY. 

5,  Define  the  intersection  of  two  tines. 

The  place  where  two  lines  meet  or  cut  each  other  is  also 
called  their  intersection. 

The  intersection  of  two  lines  is  always  a  point. 

For  example  :  the  edges  of  the  cube  intersect  at  the  cor- 
ners ;  two  lines  on  paper  which  meet  always  meet  in  a 
point. 

6,  What  are  the  parts  of  bodies,  surfaces,  and  tines  ? 

If  we  divide  a  body  into  parts,  each  part,  however  small, 
will  also  be  a  body  having  length,  breadth,  and  thickness. 
In  this  way  we  never  could  obtain  a  part  so  small  that  it  was 
a  surface,  a  line,  or  a  point. 

The  parts  of  a  body  are  bodies;  the  parts  of  a  surface  are 
surfaces;  and  the  parts  of  a  line  are  lines. 

Therefore  the  surface  of  a  body  is  not  a  part  of  the  body. 
If  any  number  of  flat  surfaces  were  placed  one  upon  another, 
they  would  have  no  thickness,  and  would  form  but  a  single 
surface.  The  common  boundary  which  separates  water  from 
oil  resting  on  it  is  neither  water  nor  oil  nor  any  other  sub- 
stance ;  it  is  not  a  body,  but  a  surface. 

Likewise  a  line  is  not  a  part  of  a  surface,  and  a  point  is 
not  a  part  of  a  line. 

7,  How  are  points  and  tines  represented  to  the  eye  ? 

Strictly  speaking,  a  point  can- 
not be  seen,  because  it  has  no 
magnitude.  On  paper  we  repre- 
sent a  point  by  a  small  dot,  and 
name  it  by  a  letter,  as  A.  The 
dot  is  really  a  very  small  body, 
but  we  do  not  regard  its  dimen- 
sions, and  think  of  it  as  having 
position  only. 


INTRODUCTION.  7 

A  line  is  represented  by  a  narrow  mark,  and  named 
generally  by  two  letters,  one  placed  at  each  end  of  the  line. 

Surfaces  are  represented  by  their  bounding  lines,  and  solids 
by  their  bounding  surfaces. 

8.  What  is  the  path  of  a  point  ? 

When  we  move  the  point  of  a  pencil  on  paper,  it  leaves  be- 
hind a  trace  which  we  call  a  line  ;  imagine  the  point  of 
the  pencil  to  be  a  point  as  understood  in  Geometry,  and  then 
the  trace  or  path  of  the  pencil  would  be  a  geometrical  line. 
Who  has  not  observed  that  if  the  red-hot  point  of  an  iron 
rod  be  moved  rapidly  in  the  dark,  the  path  of  the  point 
appears  as  a  luminous  line  ? 

The  path  of  a  moving  point  is  a  line. 

9.  Can  you  think  of  a  body  shaped  like  a  cube  ?     And  one 
shaped  like  a  cylinder?     And  one  shaped  like  a  sphere? 

10.  Is  the  space  within  the  room  a  body?     Why? 

11.  Name  the  dimensions  of  this  room  ;  the  dimensions  of 
the  floor  ;  the  dimensions  of  one  of  the  walls. 

12.  Name  the  dimensions  of  a  pencil ;  a  cistern  ;  a  brick 
wall. 

13.  Choose  among  the  bodies  before  you  one  which  has 
two  of  its  dimensions  equal.     Which  are  they  ? 

14.  Point  out  upon  a  cube  the  intersection   of    two   sur- 
faces.    Also  the  intersection  of  two  lines. 

15.  How  many  edges  and  how  many  corners  has  a  square 
box? 

16.  In  common  language,  a  telegraph  wire  is  called  a  line. 
Is  it  a  geometrical  line  ?     What  is  it  ?     Why  ? 

17.  We  call  the  trace  of  a  pencil-point  on  paper  a  line. 
What  is  it  in  reality  ?     Why  ? 

* 

/>& 


8  LESSONS   IN    GEOMETRY. 

Lesson  4. 

1.  Divide  lines  into  two  classes.    Define  each  class. 
There  are  two  kinds  of  lines,  straight  and  curved. 

A  straight  line  has  the  same  direction  from  end  to  end. 
A  curved  line  is  a  line  whose  direction  is  continually 
changing.  Curved  lines  are  often  called  curves.  A  moving 
point  describes  a  straight  line  when  the  direction  of  its 
motion  does  not  change,  and  a  curve  when  the  direction  of 
the  motion  continually  changes.  In  Figure  12, 

AB  is  a  straight  line  ; 
~B      CD  is  a  curve  ; 

EF  is  a  broken  line  ; 
GH  is  a  composite  line. 
The  figures  on   carpets   and  on 
_j.  wall-papers     furnish     numberless 
Flo  12  examples    of    broken    and    com- 

posite lines. 
The  word  "line"  used  alone  means  straight  line. 

2.  Draw  and  name  with  letters  two  straight  lines  which 
intersect.     Also  two  straight  lines  which  do  not  intersect. 

3.  The  same  exercise  with  two  curved  lines. 

4.  Draw  three  broken  lines.     Define  a  broken  line. 

5.  Draw  three  composite  lines.     Define  a  composite  line. 

6.  What  kind  of  lines  are  the  edges  of  a  cube  ?   the  edges 
of  a  prism?   the  edges  of  a  pyramid?   the  edges  of  a  cylin- 
der ?   the  edges  of  the  room  ?   the  edges  of  a  ruler  ? 

7.  What  kind  of  lines  are  represented  by  the  spokes  of  a 
wheel?   the  tire  of  the  wheel?   a  telegraph  wire?   a  watch- 
chain?   the  letters  M,  0,  S,  T,  V,  Z?  the  digits  2,  4,  6,  8? 

8.  What  is  the  path  of  a  falling  apple?    a  ball  thrown 
straight  upwards?  a  ball  thrown  sideways  into  the  air?   the 
end  of  a  clock  hand  ?   a  raindrop?   a  fly  ing  bird? 


INTRODUCTION.  9 

9.  Classify  surfaces,  and  define  each  class. 
There  are  two  kinds  of  surfaces,  plane  and  curved. 

A  plane  surface  is  a  surface  on  which  straight  lines  can  be 
drawn  in  as  many  different  directions  as  you  please. 
Plane  surfaces  are  also  called  planes. 
A  curved  surface  is  a  surface  which  is  not  plane. 

10.  What  kind  of  surfaces  are  the  faces  of  a  cube  ?  the 
faces  of  a  prism?   the  faces  of  a  pyramid?   the  base  of  a 
cone?  the  lateral  surface  of  a  cone?  the  surface  of  a  sphere? 

11.  Classify  the  surf  aces  of  the  following  bodies  :  a  table, 
a  room,  a  hat,  a  lamp-shade,  a  glass  chimney,  a  mirror,  a 
basin,  the  water  in  the  basin,  the  ocean. 

12.  Hold  a  sheet  of  paper  so  that  its  surface  is  at  first  a 
plane,  and  then  a  curved  surface. 

13.  How  can  you  test  whether  a  surface  is  a  plane  ? 

Apply  to  the  surface  the  straight  edge  of  a  ruler  in  sev- 
eral different  directions ;  if  the  edge  always  touches  the 
surface  at  all  points,  the  surface  is  a  plane. 

Test  if  the  surface  given  you  is  a  plane  or  not. 

14.  Classify  bodies,  and  define  each  class. 
Bodies  are  of  two  kinds,  polyhedrons  and  curved  bodies. 
A  polyhedron  is  a  body  bounded  wholly  by  planes. 

A  curved  body  is  a  body  bounded  partly  or  wholly  by. 
curved  surfaces. 

15.  What  kind  of  body  is  a  cube?  a  cylinder?  a  sphere? 

16.  What  does  the  word  figure  mean  in  Geometry? 
In  Geometry  every  line  or  group  of  lines  —  in  short,  every- 
thing having  shape  or  form  —  is  called  a  figure. 

17.  What  is  Geometry  ? 

Geometry  is  the  study  of  geometrical  figures,  their  proper- 
ties, their  construction,  and  their  measurement, 


CHAPTER  II. 


STRAIGHT    LINES. 


Lesson  5. 

1.  How  many  directions  has  a  straight  line? 

A  straight  line  AB  (Fig.  13)  has  two  directions,  —  the 
direction  from  A  towards  -B,  and  the  opposite  direction  from 
B  towards  A.  If  either  direction  be  known  or  given,  the 
other  direction  is  of  course  also  known. 

NOTE. — In  Geometry  "given"  is  used  very  nearly  in  the  sense  of 
"known." 

2.  How  is  the  position  of  a  straight  line  determined  9 

Different  straight  lines  may  be 

<A B>    drawn,  all  having  the  same  direc- 

,.    tion  ;  also  different  straight  lines 

^ >    may   be  drawn  through  a   given 

point  P. 

But  through  a  given  point  C  in 
a  given  direction   (as  that  of  the 
FlG  13  arrow)  only  one  straight  line  can 

be  drawn. 

Also  through  two  given  points  P  and  Q  only  one  straight 
line  can  be  drawn. 

Hence  if  we  know  either  one  point  and  the  direction  of  a 
straight  line,  or  two  points  of  the  line,  we  know  enough  about 
the  line  to  be  able  to  construct  it,  or  to  represent  it  by  draw- 
ing it  on  paper. 

In  other  words,  the  line  is  determined  in  position. 


STRAIGHT   LINES.  11 

3.  How  are  straight  lines  drawn  on  paper? 
Straight  lines  are  drawn  on  paper  by  means  of  a  ruler  with 

a  straight  edge.    -To  draw  a  line  through  two  given  points, 

lay  the  ruler  on  the  paper  so  that 

its   edge    lust   touches    the   two  ^°  ^ 

FIG. 14. 

points  ;  then  draw  the  line  with 

a  well-sharpened  pencil,  holding  the  pencil  nearly  vertical, 

and  always  touching  the  edge  of  the  ruler. 

4.  What  is  freehand  drawing  ? 

If  we  draw  a  straight  line  without  the  aid  of  a  ruler  to 
guide  the  hand,  we  are  said  to  draw  it  freehand. 

In  general,  freehand  drawing  is  drawing  without  the  aid 
of  instruments,  except  the  pencil  or  pen. 

5.  Draw  four  lines  through  a  point,  and  name  them. 

6.  Make  two  points,  and  join  them  by  a  straight  line. 
NOTE.  —  To  "  join  AB  "  means  to  "  draw  a  straight  line  from  A  to  B" 

7.  Make  three  points,  and  join  them  by  drawing  as  many 
straight  lines  through  them  as  possible. 

8.  The  same  exercise  with  four  points. 

9.  The  same  exercise  with  five  points. 

10.  Draw  freehand  a  straight  line ;   correct  or  rectify  it 
with  the  aid  of  a  ruler.     Repeat  several  times. 

11.  Draw  one  of  the  faces  of  the  body  given  you  in  three 
different  ways,  as  follows  : 

(1)  Lay  the  face  on  paper  and  trace  its  boundary. 

(2)  Lay  the  face  on  paper  ;  mark  the  corners  ;  join  them. 

(3)  Draw  the  face  as  well  as  you  can  freehand. 

12.  Test  whether  the  edge  of  your  ruler  is  straight. 

To  do  this  draw  a  line  through  two  points,  turn  the  ruler 
end  for  end,  draw  a  line  through  the  same  points  along  the 
same  edge  as  before.  The  lines  ought  to  coincide. 


12  LESSONS    IN  GEOMETKY. 

Lesson  6. 

1.  How  cure  straight  lines  and  plane  surfaces  classi- 
fied with  respect  to  the  surface  of  the  earth? 

With  respect  to  the  surface  of  the  earth  straight  lines  and 
planes  are  either  vertical,  horizontal,  or  inclined. 

A  vertical  line  is  a  line  having  the  direction  of  a  plumb 
line,  or  string  held  at  rest  in  the  hand,  and  sup- 
porting at  its  lower  end  a  small  weight  (Fig.  15). 
Every  plane  in  which  a  vertical  line  can   be 
drawn  is  a  vertical  plane. 

A  horizontal  line  is  a  line  having  the  direction 
of  a  pencil,  stick,  or  other  object  which  is  float- 
ing on  the  surface  of  still  water. 

The  surface  of  still  water,  and  every  plane  sim- 
FIG  is       ilarly  placed  with  respect  to  the  earth's  surface, 

is  a  horizontal  plane. 

A  line  or  a  plane  which  is  neither  vertical  nor  horizontal 
is  said  to  be  inclined. 

2.  Give  examples  of  lines  and  planes  vertical,  horizontal, 
and  inclined. 

3.  Hold  a  pencil  vertical,  horizontal,  inclined. 

4.  Hold  a  book  vertical,  horizontal,  inclined. 

5.  Draw  on  the  blackboard  a  line  of  each  kind. 

6.  What  direction  has  the  mast  of  a  ship  ?  the  path  of  a 
falling  apple  ?  a  rail  on  a  railway  track  ?  the  side  of  a  house  ? 
the  roof  of  the  house  ?  the  trunk  of  a  growing  tree  ? 

7.  Place  the  body  given  you  on  the  table,  and  then  exam- 
ine the  position  of  its  edges  and  its  faces  with  respect  to  the 
surface  of  the  earth. 

8.  At  what  time  of  day  are  the  sun's  rays  most  nearly 
vertical?    When  are  they  horizontal? 


STRAIGHT   LINES. 


13 


9.   When  is  the  hour-hand  of  a  clock  vertical  ?  horizontal  ? 

10.  When  two  vertical  planes  intersect,  what  kind  of  a 
line  is  the  intersection  ?     Example  :  the  walls  of  a  room. 

11.  When  a  vertical  plane  intersects  a  horizontal  plane, 
what  kind  of  a  line  is  the  intersection?     Give  an  example. 

12.  How  are  vertical  and   horizontal  lines  repre- 
sented on  paper,  when  they  are  situated  in  a  vertical 
plane  which  stands  directly  in  front  of  the  eye  ? 

Vertical  lines  are  represented  by  lines  drawn  on  paper 
straight  towards  us  or  straight  from  us  ;  horizontal  lines  by 
lines  drawn  from  left  to  right  or  from  right  to  left. 

13.  Draw  a  vertical  line,  mark  on  it  four  points,  and  draw 
through  each  point  a  horizontal  line.     Draw  freehand. 

14.  Draw  a  horizontal  line,  mark  on  it  four  points,  and 
draw  through  each  point  a  vertical  line.     Draw  freehand. 

15.  Draw  an  inclined  line,  mark  on  it  four  points,  and  draw 
through  each  point,  freehand,  a  horizontal  and  a  vertical. 

N. 


horizontal 


N.W. 


Wr 


N.E. 


-E. 


S.E. 


FIG.  16. 


FIG.  17. 


16.  How  are  the  cardinal  directions  of  a  horizontal 
plane  represented  on  paper  (Fig.  17)? 

The  cardinal  directions  are  North,  South,  East,  and  West. 
A  line  from  north  to  south  is  drawn  straight  towards  us.  A 
line  from  east  to  west  is  drawn  from  right  to  left. 


14 


LESSONS   IN   GEOMETRY. 


17.  Draw,  freehand,  through  a  point,  the  four  cardinal 
directions,  and  also  the  four  directions  which  lie  midway 
between  them  and  are  termed,  northeast,  northwest,  south- 
east, southwest.     Name  the  directions  (as  in  Fig.   17),  by 
affixing  the  letters 

N.,  S.,  E.,  W.,  N.E.,  N.W.,  S.E.,  S.W. 

18.  What  direction  is  opposite  to  N.E.  ?  opposite  to  S.E.  ? 

19.  How  many  horizontal  lines  can  be  drawn  through  a 
point?      How   many   vertical   lines?      How   many   inclined 
lines  ? 

20.  How  many  horizontal  lines  can  be  drawn  in  a  hori- 
zontal plane  ?     How  many  in  a  vertical  plane  ? 

21.  How  many  vertical  lines  can  be  drawn  in  a  vertical 
plane?     How  many  in  a  horizontal  plane? 

NOTE.  —  The  plumb  line  is  used  for  testing  vertical  walls,  and  the 
spirit  level  for  testing  horizontal  planes.  On  the  ground  the  difference 
in  level  of  two  places  is  often  found  by  means  of  a  water  level  (Fig.  18). 
It  is  a  metal  tube  mounted  horizontally  on  a  tripod  stand  with  glass 
vials  cemented  vertically  at  its  ends.  It  contains  water  colored  red  to 
distinguish  it  from  the  glass. 


FIG.  18. 


STRAIGHT   LINES. 


15 


Lesson  7. 

1.  Explain  how  a  straight  line  may  change  in 
direction. 

The  direction  of  a  straight  line  may  be  conceived  to 
change.  The  hands  of  a  clock  represent  straight  lines  which 
are  continually  changing  in  direction. 

Suppose  a  line  OA  (Fig.  19)  to  turn  about  the  point  O 
from  the  position  OA  to  the 
position  OB ;  then  the  line  is 
said  to  rotate,  or  revolve,  about 
the  point  0.  If  the  motion  con- 
tinue, the  line  will,  after  a  time, 
return  to  its  first  position  OA. 
It  is  then  said  to  have  made  one 
revolution, 

If,    during    the    motion,    the 
length    of    the    line    does    not 
change,  the  point  A  will  describe  a  curve  which  returns  into 
itself,  and  every  point  of  which  is  equally  distant  from  0. 

2.  Define  the  terms  circumference  and  circle. 

A  circumference  is  a  curve,  every  point  of  which  is  equally 
distant  from  a  fixed  point  called  the  centre. 

A  circle  is  a  portion  of  a  plane  surface  bounded  by  a 
circumference. 

3.  Define  the  terms  radius,  arc,  chord,  diameter. 

A  straight  line  drawn  from  the  centre  of  a  circle  to  the 
circumference  is  called  a  radius  (plural  radii] . 
A  part  of  a  circumference  is  called  an  arc, 
A  straight  line  joining  the  ends  of  an  arc  is  called  a  chord. 
A  chord  passing  through  the  centre  is  called  a  diameter. 


FIG. 19. 


16 


LESSONS    IN   GEOMETHY. 


4.  How  are  circles  drawn  or  described  on  paper? 
Circles,  or  strictly  speaking,  circumferences,  are  described 
on  paper  with  the  aid  of  the  compasses,  or  dividers. 

This  instrument  has  two  legs  joined 
together  by  a  pivot,  about  which  they 
can  turn.  In  the  form  best  adapted 
for  school  work,  one  leg  is  provided 
with  a  metal  point,  and  the  other  leg 
with  a  pencil  point.  In  order  to  de- 
scribe a  circle  on  paper,  we  open  the  legs 
and  proceed  as  illustrated  in  Fig.  20. 

On  the  ground  circles  are  made  as 
FIG.  20.  illustrated  in  Fig.  21. 

NOTE.  —  The  shorter  word,  "circle,"  is  often  used  instead  of  the 
word  "  circumference,"  in  cases  where  a  misunderstanding  is  not  possi- 
ble. For  example  :  "  describe  a  circle  " ;  "  an  arc  of  a  circle." 


i 


FIG.  21. 

5.  Name  in  Fig.  19  a  radius,  an  arc,  a  chord,  a  diameter. 

6.  Give  an  example  of  a  circle  or  a  circumference. 

7.  What  does  the  tire  of  a  carriage  wheel  represent?  the 
spokes  ?  the  axle  ?  the  part  of  the  tire  between  two  spokes  ? 

8.  What  is  true  of  all  radii  of  the  same  circle? 

9.  What  is  true  of  all  diameters  of  the  same  circle? 

10.    Compare  a  radius  and  a  diameter  of  the  same  circle. 


STRAIGHT   LINES.  17 

11.  Draw  a  circle,  a  radius,  a  diameter,  and  a  chord. 

12.  Describe  an  arc ;  draw  its  chord  and  radii  to  its  ends. 

13.  Describe  a  circle  ;  then  draw  (1)  a  chord  equal  to  the 
radius,  (2)  the  longest  possible  chord. 

14.  Describe  several  circles  freehand. 

15.  Describe  three  concentric  circles,  or  circles  having  the 
same  point  for  centre. 

16.  Describe  two  circles  so  that  their  circumferences  shall 
touch  each  other,  and  one  shall  lie  wholly  outside  the  other. 

17.  Describe  two  circles  so  that  their  circumferences  shall 
touch  each  other,  and  one  shall  be  wholly  within  the  other. 

18.  Describe  two  equal  circles  so  that  the  circumference 
of  each  shall  pass  through  the  centre  of  the  other. 

19.  Describe  three  unequal  circles  so  that  the  centre  of 
each  shall  lie  on  the  circumference  of  one  of  the  others. 

20.  Construct  Figure  22.      First  describe  the  circle  ;  then 
the  arcs  whose  centres  are  marked  1,  2,  3,  4,  5,  6  ;  then  find 
the  centres  of  the  remaining  arcs  by  repeated  trial. 

4 


FIG.  23. 


21 .    Construct  Figure  23.    The  circles  have  the  same  radius. 
Begin  by  describing  the  circle  in  the  middle  of  the  figure. 


18  LESSONS   IN  GEOMETRY. 

Lesson  8. 

1.  Define  parallel  lines,  and  give  examples. 
Parallel  lines  are  lines  having  the  same  direction. 

A     B      Examples.  — The  lines  AB,  CD, 

_D  EF,  and  GH  (Fig.  24)  ;    the  legs 

of  a  table  ;  the  rails  of  a  railroad 
*-  -F   (Fig.  25). 

G —  — H      Abbreviation.  —  The    sign   ||   is 

used    for    the    word   "parallel"; 
"AB  is  ||  to  CD,"  is  read  " AB  is  parallel  to  CD." 


FIG.  25. 

2.  Parallel  lines  cannot  meet  however  far  produced. 
Show  why  this  follows  from  the  definition. 

Suppose  that  two  parallel  lines,  for  example,  AB  and  CD 
(Fig.  24),  when  produced  (prolonged),  should  at  length 
meet,  and  call  the  point  of  meeting  P.  Then  we  should 
have  two  lines,  AB  and  CZ>,  drawn  through  P  in  the  same 
direction.  Now,  two  lines  so  drawn  must  coincide  (p.  10, 
No.  2).  But  AB  and  CD  plainly  do  not  coincide,  therefore 
we  know  that  they  could  never  meet,  however  far  produced. 

3.  Are  two  lines,  which  would  never  meet  if  produced, 
necessarily  parallel?     Illustrate  by  an  example. 


STRAIGHT   LINES. 


19 


4.  Problem.  —  To  draw  a  line  parallel  to  a  given 
line  AB  through  a  given  point  C. 

METHOD  I.  (Fig.  26).  —  Instruments  :  ruler  and  compasses. 

Join  C  to  any  point  D  in  AB. 

With  centre  D  and  radius  DC  describe  an  arc,  cutting 
AB  in  E. 

With  centre  C  and  radius  DC  describe  the  arc  DF. 

With  centre  D  and  radius  CE  cut  the  arc  DF in.F. 

Through  C  and  F  draw  a  straight  line. 

The  straight  line  CF  is  the  parallel  required. 


n 


E 


D 
FIG.  27. 


B 


METHOD  II.  (Fig.  27).  — Instruments  :  ruler  and  compasses. 
With  any  point  D  in  AB  as  centre  and  DC  as  radius, 
describe  a  semicircle  AECB. 

With  centre  A  and  radius  BC  cut  the  semicircle  in  E. 
Draw  CE.     The  line  CE  is  the  parallel  required. 

METHOD  III.  (Fig.  28) .  —  Instruments  :  ruler,  and  a  piece 
of  wood  with  three  straight  edges,  called  a  triangle. 

Place  the  ruler  and  triangle 
as  shown  in  the  figure.  Slide 
the  triangle  along  till  its  edge 
touches  the  point  C;  then 
draw  a  straight  line  along  the 
edge  of  the  triangle.  This 
line  will  be  parallel  to  AB 
because  the  edge  of  the  tri- 
angle during  the  motion  re- 
mains parallel  to  AB.  FIG.  28. 


20  LESSONS   IN   GEOMETRY. 

5.  Draw  a  straight  line  ;  then  draw  a  parallel  line  through 
a  point  not  in  the  line  by  Method  I. 

6.  The  same  exercise,  using  Method  II. 

7.  The  same  exercise,  using  Method  III. 

8.  Draw  a  parallel  to  a  given  line  through  a  given  point 
by  Method  I.    Test  the  accuracy  of  your  result  by  Method  III. 

9.  Draw  by  Method  III.  six  parallels  as  nearly  equidistant 
as  you  can. 

10.  Draw  a  vertical  line,  mark  five  points  on  it,  and  draw 
through  the  points  parallel  lines  by  Method  III. 

11.  Draw  six  parallel  lines,  two  vertical,  two  horizontal, 
and  two  inclined. 

12.  Draw  a  three-sided  figure,  and  then  draw  through  each 
of  its  corners  a  line  ||  to  the  opposite  side. 

13.  Describe  a  circle,  and  draw  two  parallel  chords. 

14.  Draw  a  straight  line ;   then  try  to  draw  freehand  a 
parallel  to  it.     Begin  by  marking  points  which,  as  well  as 
you  can  judge,  must  be  in  the  required  line. 

15.  Draw  freehand  a  series  of  six  parallel  lines. 

16.  Hold  two  pencils  (1)  parallel;  (2)  so  that  they  would 
intersect  if  prolonged  ;   (3)  so  that  they  are  not  parallel,  and 
also  would  not  intersect. 

17.  Point  out  on  the  bod}'  given  to  you  edges  which  are 
(1)  parallel;    (2)  intersecting;    (3)  neither  parallel  nor  in- 
tersecting. 

18.  The  same  exercise  with  the  edges  of  the  room. 

19.  When  is  a  straight  line  parallel  to  a  plane? 

A  straight  line  is  parallel  to  a  plane  when  the  line  will 
never  meet  the  plane  however  far  they  are  both  produced. 

20.  When  are  two  planes  parallel  to  each  other? 

21.  Give  an  example  of  a  line  ||  to  a  plane ;  also  of  two 
parallel  planes. 


STRAIGHT    LINES.  21 


Lesson  9. 

1.  What  has  a  straight  line  besides  direction? 
Besides  direction,  a  straight  line  has  length.     As  regards 

length,  two  straight -lines  are  either  equal  or  unequal. 

2.  Define  equal  and  unequal  straight  lines. 

Two  straight  lines  are  equal  if  they  can  be  so  placed,  one 
upon  the  other,  that  their  ends  coincide.  If  this  cannot  be 
done,  the  lines  are  unequal. 

3.  How  is  the  equality  of  two  lines  expressed  ? 
The  equality  of  two  lines  AB  and  CD  is  expressed  thus  : 

AB  =  CD.  A-  -B 

This  is  called  an  equation,  and  is  C —  —  D 

read,  "  AB  is  equal  to  CD."  E _F 

The  sign  =  is  the  sign  of  equality.  FIG.  29. 

4.  How  is  the  inequality  of  two  lines  expressed  ? 
The  inequality  of  the  two  lines  AB,  EF  is  thus  expressed  : 

AB>EF,  orEF<AB. 

These  expressions  are  read,  "  AB  is  greater  than  EF," 
and  "EF  is  less  than  AB." 

5.  How  is  the  equality  of  two  lines  tested  ? 
The  lines  are  usually  compared  with  a  third  line. 
Suppose  I  wish  to  test  whether  the  lines  AB  and  CD 

(Fig.  29)  are  equal,  I  open  the  dividers,  place  one  point  on 
A,  and  the  other  on  B.  This  is  called  "  taking  the  distance 
AB  between  the  points  of  the  dividers."  Then,  keeping  the 
opening  of  the  dividers  unchanged,  I  place  one  point  on  (7, 
and  observe  whether  the  other  point  will  fall  on  D.  If  it 
does  fall  on  Z),  I  know  that  AB  =  CD. 

Here  the  third  line  with  which  AB  and  CD  are  compared 
is  the  distance  between  the  points  of  the  dividers. 


22 


LESSONS   IN    GEOMETRY. 


6.  Draw  freehand  from  a  point  two  lines  as  nearly  equal 
as  }-ou  can,  then  test  their  equality  with  the  dividers. 

7.  The  same  exercise,  one  line  to  be  horizontal,  the  other 
vertical. 

8.  Draw  freehand  a  three-sided  figure,  with  its  sides  as 
nearly  equal  as  you  can  ;  test  the  equality  with  the  dividers. 

9.  Draw  four  parallels  exactly  equal  in  length. 

10.  Draw  a  line  equal  to  an  edge  of  the  body  given  you. 

11.  Draw  lines  a,  &,  c,  d  so  that  a  =  6,  c  >a,  d  <  b. 

12.  Read  the  following  :  m  =  n,  AB  <  CZ>,  x  >  y. 

13.  Draw  four  straight  lines,  a,  6,  c,  d ;  then  compare  a 
with  each  of  the  other  lines,  writing  the  result  with  the  proper 
sign. 

14.  Draw  freehand  two  lines  as  nearly  equal  as  you  can, 
each  longer  than  the  greatest  opening  of  your  dividers.    Can 
you  test  their  equality  by  means  of  your  dividers  ? 

15.  Compare    different    lines    drawn    between  two 
points. 

A  straight  line  is  the  shortest  line  between  two  points ; 
hence  the  length  of  a  straight  line  joining  two  points  is  taken 
as  the  distance  of  the  points  from  each  other. 

rr 


FIG.  30. 


16.    How  do  sign-painters  make  use  of  this  truth  (Fig.  30)  ? 

They  chalk  a  cord  and  stretch  it  tightly  between  the  points 

through  which  the  line  is  to  pass  ;  then,  seizing  this  cord  by 


STRAIGHT   LINES.  23 

the  middle,  they  draw  it  back  a  little  from  the  wood,  and 
then  let  it  go.  It  springs  back,  strikes  the  wood  a  sharp 
blow,  and  leaves  -on  it  a  white  trace,  which  is  a  straight  line. 

17.  Define  an  axiom,  and  illustrate  the  meaning. 
There  are  statements  or  assertions  which  are  so  obvious 

that  they  stand  in  no  need  of  any  explanation  or  proof. 
Every  one  sees  at  a  glance  that  they  are  true. 

In  Geometry,  an  assertion  which  is  admitted  to  be  true, 
without  proof,  is  called  an  axiom. 

An  instance  of  an  axiom  occurs  in  No.  5.  We  assume 
that  if  AB  and  CD  are  separately  equal  to  the  distance  be- 
tween the  points  of  the  dividers,  they  are  equal  to  each  other. 

In  general,  if  any  two  magnitudes  are  each  equal  to  a  third 
magnitude,  we  assume,  as  quite  obvious,  that  they  are  equal 
to  each  other. 

Another  example  of  an  axiom  is  the  assertion  that  a 
straight  line  is  the  shortest  distance  between  two  points. 

18.  State  the  most   important  axioms  which  are 
found  useful  in  the  study  of  Geometry. 

1.  Two  magnitudes,  each  equal  to  a  third,  are  equal  to  each 
other;  or,  for  any  magnitude,  its  equal  may  be  substituted. 

2.  If  equals  are  added  to  equals,  the  sums  are  equal. 

3.  If  equals  are  taken  from  equals,  the  remainders  are 
equal. 

4.  If  equals  are  multiplied  by  equals,  the  products  are  equal. 

5.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

6.  A  ivhole  is  greater  than  any  of  its  parts. 

7.  Through  a  given  point  in  a  given  direction,  only  one 
straight  line  can  be  drawn. 

8.  Through  two  points,  only  one  stmight  line  can  be  drawn. 

9.  A  straight  line  is  the  shortest  dis&mee  between  two  points. 

10.  Through  a  given  point  only  one  parallel  to  a  given 
straight  line  can  be  drawn* 


LESSONS   IN    GEOMETRY. 


Lesson  1C. 

1.  Explain  the  addition  and  subtraction  of  lines. 

In  order  to  add  two  straight  lines,  let  AB  and- CD  (Fig. 
31)  be  two  lines,  draw  a  straight  line  and  take  on  it 
EF=  AB,  FG=CD\  then  the  line  EG  will  be  the  sum  of 
AB  and  BC ;  or, 

EG  =  AB  +  CD. 

A\—  —IB  C\ \D 


El 


FIG.  31. 


If,  instead  of  laying  off  CD  towards  the  right,  we  lay  it  off 
from  F  to  H  towards  the  left,  then  we  obtain  a  length  EH 
which  is  the  difference  between  AB  and  CD  ;  or,  using  signs, 

EH=AB-CD. 

2.  How  are  lines  multiplied  or  divided  by  numbers  ? 

To  multiply  a  line  by  a  number  is  to  add  the  line  repeat- 
edly to  itself.  Thus  :  produce  AB  (Fig.  32)  by  adding  to 
it  lines  BC,  CD,  etc.,  each  equal  to  itself  ;  then  AC  is  twice 
as  long  as  AB,  AD  three  times  as  long,  etc. 

A,  _  L*  _  ^  _  L5  _  ^  _  ,£  _  ^  _  Jl  _  lK_tL 

FIG.  32. 

These  results  are  usually  written  as  follows  : 
AC=2AB,  AD  =  SAB,  AE  =  ±AB,  AL  =  9AB. 

Division  is  the  inverse  of  multiplication.  To  divide  a  line 
by  a  number  is  to  separate  the  line  into  as  many  equal  parts 
as  there  are  units  in  the  number,  and  take  one  of  the  parts 
as  the  quotient.  The  operation  of  division  is  usually  ex- 
pressed in  the  form  of  a  fraction,  thus  (see  Fig.  32)  : 


ete. 


STRAIGHT   LINES. 


25 


3.  Problem.  —  To  bisect  a  given  straight  line  AB. 
With  centres  A  and  B  (Fig.  33) ,  and  a  radius  greater  than 
half  of  AB,  describe  arcs  intersecting  at  C  and  D. 
Join  CD,  which  will  cut  AB  at  a  point  M. 
Then  M  is  the  middle  point  of  AB. 


H 


K 


FIG.  34. 


4.  Problem.  —  To  divide  a  given  straight  line 
into  any  number  (say  five)  equal  parts. 

Draw  through  A  (Fig.  34)  any  straight  line  AX. 
Upon  AX  lay  off  five  equal  lengths  from  A  to  G. 
Join  GB,  and  draw  through  the  points  C,  D,  E,  F,  between 
A  and  G,  lines  ||  to  GB  (p.  19,  No.  4). 

These  parallels  will  divide  AB  into  five  equal  parts. 

NOTE. —  The  use  of  the  lines  CM,  DN,  etc.,  will  appear  hereafter. 

5.  Draw  any  two  straight   lines ;    then  construct  a  line 
equal  to  their  sum,  and  a  line  equal  to  their  difference. 

6.  In  Fig.  32  name  lengths  equal  to  the  following : 

AE  +  BD,  AC  +  GL,  AE  -  AD,  AG  -  EH,   AD  -  HL. 

± AC,  2x  (AB  +  BD),  3  x  (DH-DE),  ±x(AG-BF). 

±AE,  IAL,  \BL,  ±(AD  +  AH),  ±(CL-CF). 

7.  Draw  a  broken  line,  then  develop  it ;  that  is,  draw  a 
straight  line  equal  to  the  entire  length  of  the  broken  line. 

8.  Draw  a  straight  line  equal  in  length  to  the  sum  of  the 
edges  of  one  face  of  the  solid  given  to  you. 


26  LESSONS   IN    GEOMETRY. 

9.  Draw  AB  =  CD,  and  MN=  PQ.      Then  draw  lines 
equal  to  AB  +  MN,  CD  +  PQ,  AB  -  MN,  CD  -  PQ. 

Are  the  first  two  of  these  lines  equal?     Why?     Are  the 
last  two  of  these  lines  equal?     Why? 

10.  Produce  a  line  AB  to  P,  making  AP=  5  AB. 

11.  Draw  a  line  AB  and  then  bisect  it. 

12.  Divide  a  line  into  four  equal  parts.     First  bisect  the 
line  and  then  bisect  each  of  the  halves. 

13.  Divide  a  line  into  six  equal  parts. 

14.  Divide  a  line  into  seven  equal  parts. 

15.  Bisect  a  line  freehand.     Test  and  correct  the  result. 

16.  Trisect  a  line  freehand.     Test  and  correct  the  result. 


D 


A  F  B  A.  E  B 

FIG.  35.  FIG.  36. 

17.  Bisect  a  line  by  repeated  trial  with  the  dividers. 

18.  Draw  a  three-sided  figure  ABC.     Bisect  its  sides  at 

D,  E,  F.     Join  the  middle  points  to  the  opposite  corners. 
If  your  work  is  accurate,  the  lines  AD,  BE,  CF  will  all  pass 
through  the  same  point  0  (Fig.  35). 

19.  Draw  a  four-sided  figure  ABCD.     Bisect  its  sides  at 

E,  F,  G,  H.     Join  the  middle  points,  taken  in  order  around 
the  figure.     If  your  work  is  accurate,  EF  will  be  ||  to  GH, 
and  FGl  to  EH  (Fig.  36). 

NOTE.  —  In  performing  Exercises  18  and  19  do  not  try  to  choose 
points  so  that  your  figures  will  resemble  Figures  35  and  36,  but  mark 
their  positions  on  your  paper  without  reference  to  these  figures.  Do 
not  choose  your  points  too  near  together ;  the  farther  they  are  from 
each  other  the  better. 


STKAIGHT   LINES.  27 


Lesson  11. 

1.  What  is  a  unit  of  length?   Give  examples. 

A  unit  of  length  is  a  length  named  and  defined  by  law. 
Examples  :  the/oo£,  the  yard,  the  mile,  the  meter, 

2.  What  is  meant  by  measuring  a  line? 

To  measure  a  line  is  to  find  how  many  times  it  will  con- 
tain a  unit  of  length.  The  number  of  times  a  line  will  con- 
tain a  unit  of  length,  joined  to  the  name  of  the  unit,  is  called 
the  length  of  the  line.  Examples  :  6  feet,  9  miles. 

3.  What  units  of  length  are  in  common  use? 

The  units  of  length  in  common  use  in  this  country  and  in 
England  are  the  inch,  the  foot,  the  yard,  the  rod,  the  mile. 
Abbreviations  :  in.  =  inch,  ft.  =  foot,  yd.  =  yard. 

4.  How  are  these  units  related  to  one  another? 
They  are  related  as  shown  in  the  following  table : 

12  inches  =  1  foot.  5|-  yards  =  1  rod. 

3  feet     =  1  yard.          320    rods    =  1  mile. 

5.  How  are  lengths  less  than  an  inch  expressed? 
The  inch  is  usually  divided  into  halves,  quarters,  eighths, 

and  sixteenths.  If  we  wish  to  take  into  account  very  small 
parts  of  an  inch,  it  is  more  convenient  to  divide  it  decimally 
into  tenths,  hundredths,  etc. 

6.  How  many  inches  in  one  rod?    in  one  mile? 

7.  How  many  feet  are  there  in  one  mile? 

8.  How  many  inches  in  3  feet?  10  feet?  21  yds.? 

9.  What  part  of  a  foot  is  3  inches?  4  inches?  8  inches? 
9  inches?  10  inches? 

10.  Reduce  100  inches  to  feet.     Also  to  yards. 


28  LESSONS   IN   GEOMETRY. 

11.  You  have  a  scale  for  measuring  lengths.    What  is  the 
length  of  the  scale  ?     Describe  how  each  inch  is  subdivided. 
Name  one  of  the  shortest  parts.     Name  two  of  them  taken 
together.     Name  three  of  them  taken  together,  etc. 

12.  How  many  half-inches  are  there  in  5^  inches? 

13.  How  many  quarter-inches  are  there  in  4|  inches? 

14.  Reduce  9J  inches  to  quarters  of  an  inch. 

15.  Reduce  3^  inches  to  quarters  of  an  inch. 

16.  Reduce  2J  inches  to  eighths  of  an  inch. 

17.  Reduce  4J  inches  to  sixteenths  of  an  inch. 

18.  Reduce  24  quarters  of  an  inch  to  inches. 

19.  Reduce  28  eighths  of  an  inch  to  inches. 

20.  Find  the  difference  between  6J  in.  and  6^|  in. 

21.  How  many  eighths  of  an  inch  are  there  in  one  foot? 

22.  How  many  quarter-inches  are  there  in  one  yard? 

23.  Draw  straight  lines  having  lengths  as  follows  : 

4fin.  ;   3f  in.  ;  2|  in.  ;  1ft  in. 

24.  Through  a  point  A  draw  three  straight  lines.     Upon 
one  of  them,  mark  off  a  length  AB  =  2|  inches  ;  upon  an- 
other, AC—  3£  inches  ;  upon  the  other,  AD  =  4T3g-  inches. 

25.  Draw  four  parallel  lines   (7,  4).     Then  lay  off  upon 
the  first  AB  =  4J  in.  ;  upon  the  second,  CD  =  2J  in.  ;  upon 
the  third,  EF  =  AB  +  CD',   upon  the  fourth,   GH=AB- 
CD.     Then  measure  EF  and  GH.     Do  not  put  the  scale  on 
the  paper.     Record  the  lengths  of  EF  and  GH  thus  : 

EF=          ,  GH= 

26.  Draw  a  broken  line,  ABCDEF,  making  AB=  2  in., 
BC=  2f    in.,    CD  =  l±  in.,    DE  =  Z^   in.,    EF=2%  in. 
Then  develop  it.     In  what  part  does  the  middle  point  of  the 
developed  line  lie  ?      Find  the  length  of  the  developed  line 
(1)  by  adding  the  parts,  (2)  by  measuring  it. 


STRAIGHT   LINES. 


Lesson  ±2. 

DIRECTIONS    TO    BE    FOLLOWED. 

Work  out  Exercises  1  and  2  before  the  hour  of  recitation,  and  bring 
the  results  with  you.  Do  each  of  the  other  exercises  twice;  first,  using 
no  instruments  but  your  pencil,  and  estimating  all  lengths  as  well  as  you 
can  by  your  eye ;  secondly,  using  rule,  scale,  and  dividers,  and  measur- 
ing lengths  correct  to  a  sixteenth  of  an  inch  at  least.  Record  neatly  all 
your  results,  both  the  estimated  lengths  and  the  measured  lengths. 

1.  Find  the  average  value  of  your  pace.     To  do  this,  take 
ten   steps   straight   forward  as   you   naturally  walk,    meas- 
ure the  distance  walked,  and  divide   this   distance  by  ten. 
Repeat,  and  if  the  two  results  differ,  take  their  mean,  which 
is  equal  to  half  their  sum. 

2.  Measure  by  pacing  the  distance  from  the  schoolhouse 
to  3*our  home. 

3.  Draw  a  three-sided  figure  ABC.     Then   measure  and 
record  the  lengths  of  the  sides,  AB,  BC,  AC. 

FORM    OF    RECORD. 


AB 

BC 

AC 

Estimated  Length. 
Measured  Length. 

Difference. 

4.  Measure  the  length  (1)  of  one  of  the  horizontal  bars 
in  the  form  of  record  above,  (2)  of  one  of  the  vertical  bars. 
Make  a  neat  form  of  record  for  yourself. 

5.  Draw  AB=  1|  in.,  AC  =  2   in.     Join   BC.     Produce 
AB  to  D  making  ED  =\\  in.     Draw  DE  II  to  BC  and  meet- 
ing AC  produced  at  E.     Measure  BC,  DE,  CE. 

6.  Measure  and  record  the  dimensions  of  the  body  which 
is  given  you. 


30  LESSONS   IN   GEOMETRY. 


Lesson  13. 

1.  Suppose  three  points  A,  B,  C  on  the  ground  are 
in  a  straight  line,  and  that  by  measurement  you  find 
AB  =  1%5  feet,  AC  =300  feet.    How  can  these  lengths 
be  represented  on  this  page  by  a  straight  line  drawn 
from  left  to  right  ? 

Sincfc  the  width  of  the  printed  matter  on  the  page  is  only 
about  31  inches,  it  is  clear  that  the  line  which  I  draw  to  rep- 
resent AC  cannot  be  more  than  31  inches  long.  Suppose  I 
take  1  inch  on  paper  to  represent  100  feet  or  1200  inches  on 
the  ground.  Then  T^7  of  an  inch  will  represent  1  foot. 
Therefore  300  feet  will  be  represented  by  f$$  inches,  or  3 
inches,  and  125  feet  by  ij|-  inches,  or  11  inches.  Hence, 
to  represent  the  measured  lines  I  draw  on  the  paper  AC '  =  3 
inches,  and  take  AB  =  1 J  inches. 

A  B  C 

_j | L_ 

Scale:  1  in.  =  100  ft. 

The  two  measured  lengths  have  now  been  drawn  to  scale; 
the  reducing  factor  is  1200  ;  the  scale  of  reduction  is  y^Vo'  or 
as  sometimes  written  1  :  1 200  ;  a  common  way  of  expressing 
it  is  as  follows  :  1  inch  =  100  feet. 

2.  What  is  meant  by  "  drawing  to  scale  "  ? 

Drawing  to  scale  means  drawing  lines  on  paper  so  that 
each  line  shall  be  the  same  fractional  part  of  the  line  on 
the  ground  which  it  represents. 

3.  What  is  the  reducing  factor  ? 

The  Reducing  factor  is  the  number  by  which  each  line  on 
the  ground  must  be  divided  in  order  to  obtain  the  reduced 
length  to  lay  off  on  paper. 


STRAIGHT   LINES.  31 

4.  What  is  the  scale  of  reduction  ? 

The  Scale  of  reduction  is  the  fraction  having  1  for  nume- 
rator and  the  reducing  factor  for  denominator. 

In  making  a  drawing,  the  scale  of  reduction  should  be  so 
chosen  as  to  leave  a  fair  margin  around  the  paper  after  the 
lines  are  all  drawn,  and  should  be  neatly  written  by  the  side 
of  the  drawing,  usually  at  the  lower  right-hand  corner. 

5.  Give  examples  of  drawing  to  scale. 

On  maps  distances  of  hundreds  and  even  thousands  of 
miles  are  represented  by  lines  a  few  inches  long. 

Plans  of  estates  and  buildings  are  always  drawn  to  a 
reduced  scale.  In  general,  the  same  is  true  of  all  pictures 
and  paintings. 

6.  Write  in  the  form  of  a  fraction  the  following  scales : 
1  in.  — 1  yd.,  \  in.  =  l  ft.,  2  in.  =  l  mile,  ^  in.  =  10  miles. 

7.  If  the  scale  of  reduction  be  ^V^,  what  length  on  the 
ground  will  be  represented  by  1  inch  on  the  paper  ? 

8.  If  4  inches  on  paper  represent  a  line  200  feet  long, 
express  the  scale  of  reduction  in  a  fractional  form.     "What 
length  on  paper  will  represent  half  a  mile  ? 

9.  The  scale  of  a  map  is  -^Vo-     The  distance  between 
two  towns  on  the  map  is  8.8  inches  ;  what  is  the  real  distance 
between  the  towns  ? 

10.  Draw  5  parallels  and  lay  off  on  them  to  the  scale  -^ 
the  lengths :  100  feet,  150  feet,  70  feet,  50  feet,  30  feet. 

11.  Draw  a  three-sided  figure,  and  find  what  lengths  the 
sides  would  represent  to  the  scale  1  inch  =  1000  feet. 

12.  Find  from  a  map  of  the  United  States  the  distance  in 
a  straight  line  from  New  York  to  Chicago. 

13.  Measure  the  line  on  the  blackboard  assigned  to  you, 
and  record  its  length.     Then  represent  the  line  on  paper, 
choosing  a  suitable  scale  of  reduction  for  this  purpose. 


82  LESSONS   IN   GEOMETRY. 


Lesson  14. 

1.  What  is  the  nature  of  the  Metric  System  ? 
The  Metric  System  of  units  is  a  decimal  system^ 
The  fundamental  unit  of  length  is  the  meter. 

All  the  other  units  of  length  are  derived  from  the  meter  by 
decimal  multiplication  and  division. 

2.  Write  out  in  tabular  form  the  names,  abbrevia- 
tions, and  relative  values  of  the  units  of  length. 

THE    METRIC    UNITS    OF    LENGTH. 

Name.  Abbreviation.                       Relative  value. 

Meter m. 

Dekameter dkm 10       meters. 

Hectometer hm 100 

Kilometer km 1000 

Decimeter dm 0.1      of  a  meter. 

Centimeter  .......  cm 0.10      " 

Millimeter mm 0.100    " 

The  units  most  largely  employed  in  practice  are  the  meter, 
the  kilometer,  the  centimeter,  the  millimeter. 

3.  How  is  one  unit  reduced  to  another  unit? 
Reduction  from  a  larger  unit  to  a  smaller  unit  is  performed 

by  multiplying  by  10  as  many  times  as  may  be  necessary ; 
that  is,  by  moving  the  decimal  point  towards  the  right  the 
proper  number  of  places. 

Reduction  from  a  smaller  unit  to  a  larger  unit  is  performed 
by  dividing  by  10  as  many  times  as  may  be  necessary  ;  that 
is,  by  moving  the  decimal  point  towards  the  left  the  proper 
number  of  places. 

Examples.    32m   = 320dm  =  3200cm  =  32000mm  ; 

0.764km  =  7.64hm  =  76.4dkm  =  764m  =  764000mm; 
376mm  =  37.6cm  =  0.376m  =  0.000376km. 


STRAIGHT   LINES. 


4.  In  practice,  only  one  unit  is  used  in  expressing  a  length. 
Suppose  the  length  of  a  line  is  measured  and  found  to  be 
7dm,  6cm.     Reduce  this  to  meters  ;  decimeters  ;  centimeters. 

5.  Reduce  12.4km  to  each  of  the  other  units. 

6.  Reduce  1800cm  to  decimeters  ;  to  meters  ;  to  kilometers. 

7.  What  part  of  a  kilometer  is  1  millimeter? 

8.  What  part  of  a  meter  is  5cm?  2dm?  10mm? 

9.  41™  _|_  4™  4.  4'm  _j_  4mm  _  }1OW  many  meters  ? 

10.  What  is  the  difference  between  4em  and  28mm? 

11.  From  10m  take  10mm. 

12.  How  long  a  piece  of  wood  is  required  to  make  20 
rulers,  each  ruler  to  be  30cm  long? 

13.  How  many  rails  7.5rn  long  are  required  to  build  a  rail- 
road track  300kmlong? 

14.  Reduce  the  following  values  to  English  equivalents : 
height  of  a  barometer,  76cm ;  height  of  Mt.  Blanc,  4815m. 

15.  Reduce  236  miles  to  kiloYneters. 

16.  Describe  the  divisions  of  your  metric  scale. 

17.  Measure  to  the  nearest  millimeter  (1)  the  length  of 
this  page  ;   (2)'  the  length  of  a  line  of  print  upon  the  page. 
Record  the  results  neatly  in  a  tabular  form. 

18.  Construct  on  paper  a  scale,  ldm  long,  precisely  like 
the  decimeter  scale  printed  below. 


3 

TTTTlTTTT 


4 

TTlTTTT 


TTTT 


10 


TABLE    OF    ENGLISH    AND    METRIC    EQUIVALENTS. 

More  exactly. 

1  kilometer  =  0.6214  mile. 
1  meter  —39.37  inches. 
1  centimeter  =  0.3937  inch. 


Approximately. 
8  kilometers     =  5  miles. 

1  meter  =  1TJT  yards. 

2  centimeters  =  1  inch. 


34  LESSONS  IN   GEOMETRY. 


Lesson  15. 

1.  Express  in  fractional  form  the  scale  4CW  =  lm. 

2.  Express  in  fractional  form  the  scale  2mm  =  I*"1. 

3.  A  certain    map   is   drawn   to    the    scale    Iem=l00m. 
What  length  on  the  map  will  represent  a  distance  of  1.2km? 

4.  The  distance  between  two  towns  on  a  map  drawn  to 
the  scale  1  :  40000  is  represented  by  a  line  ldm  in  length. 
What  is  the  actual  distance  between  the  towns? 

5.  A  monument  is  40m  high,  and  a  man  standing  beside 
it  is  2m  high.      Draw  to  the  scale  1 :  200  straight  lines  to 
represent  the  monument  and  the  man. 

6.  Draw  three  parallel  lines  of  any  lengths,  and  then  find 
what  lengths  they  would  represent  to  the  scale  1  :  200.     For 
example :     a   line   3cm    long   would    represent    3cm  x  200  = 
600cm  =  6m. 

7.  Draw  AB=26mm,  AG=37mm.      Join  BC.     Produce 
AB  to  D,  making  BD  =  50mm.      Draw  DE  \\  to  BC,  and 
meeting  AC  produced  at  E.     Measure  BC,  DE,  CE. 

8.  Draw  a  line  by  your  eye  4cm  long  ;  measure  it ;  record 
your  error  in  millimeters.     Repeat  several  times. 

NOTE.  —  In  doing  Exercises  9,  10,  and  11  follow  the  directions  given 
in  Lesson  12,  and  measure  lengths  to  the  nearest  ^//-millimeter. 

9.  Measure  the  dimensions  of  the  following  figure : 


I) 


10.  Draw  a  four-sided  figure  ABCD,   and  measure  the 
lengths  of  the  sides. 

11.  Measure  the  dimensions  of  the  body  given  you. 


STRAIGHT    LINES.  S5 

Lesson  16.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  5-9. 

2.  What  is  the  greatest  possible  number  of  straight  lines 
that  can  be  drawn  through  four  points?     Mark  four  points, 
and  draw  all  the  straight  lines  possible  through  them. 

3.  Place  four  points  so  that  four  and  only  four  straight 
lines  can  be  drawn  through  them.     Draw  the  lines. 

4.  In  how  many  points  will  five  lines  intersect  if  four  of 
the  lines  are  parallel  ?  if  three  are  parallel  ?  if  two  are  par- 
allel? if  no  two  of  the  lines  are  parallel? 

5.  Through  a  vertical  line  how  many  vertical  planes  can 
be  passed  (that  is,  how  many  different  vertical  planes  can  be 
imagined,  all  of  which  shall  contain  the  vertical  line)  ?  how 
many  horizontal  planes?  how  many  inclined  planes? 

6.  Through   a  horizontal  line  how  many  vertical   planes 
can  be   passed?   how  many  horizontal   planes?   how  many 
inclined  planes? 

7.  Through   an   inclined  line   how  many  vertical   planes 
can  be  passed?   how  many  horizontal   planes?    how   many 
inclined  planes? 

8.  Describe  two  concentric  circles.      Then  describe  two 
circles  which  shall  touch,  but  not  cut,  the  concentric  circles. 

9.  Describe  a  circle  with  radius  two  inches,  and  draw  in 
it  a  chord  equal  to  one  and  one-half  times  the  radius. 

10.  Describe  a  circle,  and  within  this  circle  two  smaller 
circles  which  shall  touch  each  other  and  the  large  circle. 

11.  Describe   a   semicircle,  and  draw  in  it  three   chords 
parallel  to  the  diameter. 

12.  Give  an  example  of  parallel  lines  ;  of  parallel  planes. 

13.  Describe  the  situation  of   all  the   points  in  a  plane 
which  are  one  inch  from  a  straight  line  drawn  in  the  plane. 


86  LESSONS   IK   GEOMETRY. 

Lesson  17.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  10-13. 

2.  Which  is  the  greater,  3.6  in.  or  3|  in.  ?     What  is  the 
difference  ? 

3.  If  a  pace  =  21  ft.,  how  many  paces  are  there  in  1  mile? 

4.  Express  as  fractions  the  following  scales  :  1  in.  =  1  ft. ; 
1  in.  =  1  yd.  ;   1  in.  =  1  rod ;   1  in.  =  1  mile. 

5.  If  4  in.  represent  66  ft.,  what  will  represent  44  yds.  ? 

6.  Among  the  scales  in  use  in  the  engineer  service  are : 

Plans  of  buildings,  1  in.  —  10  ft. 

Maps  1|  miles  square,  2  ft.  =  1  mile. 

Maps  3  miles  square,  1  f t.  =  1  mile. 

Maps  4  to  8  miles  square,  6  in.  =  1  mile. 

Maps  9  miles  square,  4  in.  =  1  mile. 

Maps  12  to  24  miles  square,  2  in.=  1  mile. 

Maps  50  miles  square,  1  in.=  1  mile. 

Express  these  scales  in  the  form  of  fractions. 

7.  Divide  by  your  eye  an  inch  into  tenths. 

8.  Divide  a  line  into  8  equal  parts. 

9.  Draw  a  three-sided  figure ;    bisect  its  sides,  and  pro- 
duce (if  necessary)  the  bisecting  lines  till  they  meet. 

They  should  all  meet  in  the  same  point. 

10.  Draw  a  line  AB,  and  divide  it  into  three  parts,  such 
that  the  second  part  shall  be  double  the  first,  and  the  third 
part  double  the  second. 

11.  Draw  a  four-sided  figure,  two  sides  of  which  are  par- 
allel ;  then  describe  circles  upon  the  sides  taken  as  diameters. 

12.  Draw  a  four-sided  figure  ABCD,  having  its  opposite 
sides  parallel ;    join  AC  and  BD,  and  mark  their  point  of 
intersection   0.      Measure  the  lengths  -4(7,  BC,  CD,  DA, 
AO,  CO,  BO,  DO. 


CHAPTER  III. 


ANGLES. 


Lesson  18. 

1.  Define  an  angle,  its  sides,  and  its  vertex. 

An  angle  is  the  difference  in  direction  of  two  straight  lines 
which  meet,  or  would  meet  if  produced. 

The  lines  are  called  the  sides  of  the  angle  ;  their  point  of 
meeting  is  called  the  vertex. 

2.  Illustrate  these  definitions. 

The  lines  AB  and  AC  (Fig.  37)  form  the  angle  BAC.  A 
is  the  vertex,  AB  and  AC  the  sides.  The  lines  DE  and 
FG  (Fig.  38)  also  form  an  angle.  Its  vertex  is  the  point  H, 
where  the  lines  would  meet  if  produced. 


A  B  D  E  II 

FIG.  37.  FIG.  38. 

3,  How  is  an  angle  named  or  denoted? 
An  angle  may  be  named  in  either  of  two  ways  : 

(1)  By  a  small  letter  placed  just  inside  the  vertex. 

(2)  By  writing  the  letter  which  stands  outside  the  vertex 
between  two  letters  which  stand  one  on  each  of  the  sides. 
Thus,  the  angle  in  Fig.  37  is  the  angle  m,  or  the  angle  BAC. 

Abbreviations  :  /,  stands  for  k'  angle,"  A  for  "  angles." 


38 


LESSONS   IN   GEOMETRY. 


D. 


II 


4.  Angles  differ  in  magnitude.     How  can  a  clear 
idea  of  the  magnitude  of  an  angle  ~be  obtained  ? 

To  get  a  clear  idea  of  angular  magnitude  we  must  conceive 
an  angle  to  be  described  by  the  rotation  of  a  line.  ^ 

Suppose  a  line  to  start  from  the 
position  OA  (Fig.  39),  and  turn 
about  the  point  0  in  the  direction 
of  the  arrow.  When  the  line  has 
reached  the  position  OJ5,  it  has 
described  the  angle  AOB;  when 
it  has  reached  the  position  OCy, 
it  has  described  the  angle  AOC. 
If  the  motion  continue,  an  angle 
larger  than  AOC  will  be  described. 
It  is  quite  obvious  that  the  mag- 
nitude of  the  angle  depends  entirely  upon  the  amount  of  ro- 
tation of  the  line,  and  not  at  all  upon  the  length  of  the  line. 

5.  Define  the  right  angle  and  the  straight  angle. 
When  a  line  makes  one-fourth  of  a  complete  revolution  it 

describes  an  angle  which  is  called  a  right  angle. 

Thus,  the  angler  AOC  (Fig.  39)  is  a  right  angle. 

When  a  line  makes  one-half  of  a  complete  revolution  it 
describes  an  angle  which  is  called  a  straight  angle.  The 
angle  AOE  (Fig.  39)  is  a  straight  angle  ;  its  sides  are  oppo- 
site in  direction,  and  form  a  straight  line  ;  hence  its  name. 

6.  Can  an  angle  be  greater  than  a  straight  angle  ? 

Yes.  When  the  rotating  line  (Fig.  39)  has  made  three- 
fourths  of  a  revolution  it  takes  the  position  06r,  and  the 
angle  described,  AOG,  is  equal  to  three  right  angles.  The 
side  OG  of  this  angle  is  opposite  in  direction  to  the  line  0(7. 

When  the  rotating  line  has  made  exactly  one  revolution  it 
arrives  at  the  position  OA  from  which  it  started,  and  has 
described  an  angle  equal  to  four  right  angles. 


ANGLES.  39 

7.  Define  an  acute  angle  and  an  obtuse  angle. 

An  angle  less  than  a  right  angle  is  called  an  acute  angle, 
and  an  angle  greater  than  a  right  angle  is  called  an  obtuse 
angle. 

8.  What  kind  of   an  angle    (Fig.   39)   is  AOB?  AOD? 
DOE?  COE?  FOG?  EOG?  DOG?  GOH?  HOC? 

9.  Is   the  angle   a   on   the   blackboard   right,   acute,   or 
obtuse?  the  Z  6?  the  Z  c?  etc. 

10.  Mark  three  points  ^4,  -B,  C.     Join  them  by  straight 
lines.     How  many  angles  are  formed?     Name  them. 

11.  Draw  two  intersecting  lines.     How  many  angles  are 
formed?     Name  them,  using  as  few  letters  as  possible. 

12.  Draw  freehand  an  acute  angle,  an  obtuse  angle,  a 
right  angle. 

13.  What  angle  does  a  vertical   line    make  with  a  hori- 
zontal line  ? 

14.  What  kind  of  an  angle  do  the  hands  of  a  clock  make 
with  each  other  at  three  o'clock  ?  at  six  o'clock  ? 

15.  Mention  times  of  the  day  when  the  hands  of  a  clock 
make  a  right  angle,  an  acute  angle,  an  obtuse  angle. 

16.  How  long  a  time  does  it  take  the  minute-hand  of  a 
watch  to  describe  a  right  angle  ?     How  long  does  it  take  the 
hour-hand  to  do  the  same  ? 

17.  What  kind  of  an  angle  does  the  hour-hand  describe  in 
one  hour?  three  hours?  five  hours?  six  hours? 

18.  A  man  is  walking  due  north.     If  he  suddenly  change 
the  direction  in  which  he  is  moving  through  two  right  angles, 
in  what  direction  will  he  now  be  going?     In  what  direction 
if  he  makes  a  change  from  due  north  through  one  right  angle 
to  his  left?   through  three  right  angles  to  his  right?   through 
two  and  one-half  right  angles  to  his  right? 


40 


LESSONS   IN  GEOMETRY. 


Lesson  19. 

1.  Define  perpendicular  lines  and  oblique  lines. 

If  two  lines  form  a  right  angle,  they  are  said  to-be  perpen- 
dicular to  each  other ;  if  they  form  any  other  angle,  they  are 
said  to  be  oblique  to  each  other. 

Abbreviation  :    The  sign  _L  means  "  perpendicular." 

2.  Problem.  —  To  erect  a  perpendicular  at  any  point 
C  of  a  given  straight  line  AB. 

METHOD  I.  (Fig.  40) .  — Instruments  :  ruler  and  compasses. 
With  centre  (7,  and  any  radius,  cut  AB  in  D  and  E. 
With  centres  D  and  E,  and  a  greater  radius  than  before, 
describe  arcs  intersecting  at  F. 

Join  CF;  OF  is  the  perpendicular  required. 

D\ 


C 

FIG.  40. 


E 


B 


FIG.  41. 


METHOD  II.  (Fig.  41).  —  Instruments:  ruler  and  triangle. 
Place  the  ruler  and  the  triangle  as  seen  in  the  figure. 
Draw  a  line  along  the  edge  CD. 
This  line  will  be  the  perpendicular  required. 

3.  Give  examples  of  perpendicular  lines. 

4.  Draw  a  line  AB ;  erect  a  perpendicular  by  Method  I. 

5.  Draw  a  horizontal  line,   and  erect  (by  Method  II.) 
perpendiculars  at  six  equidistant  points  along  the  line. 

6.  How  many  right  angles  are  formed  when  two  perpen- 
dicular lines  cut  each  other? 


ANGLES.  41 

7.  Problem.  —  To  erect  a  perpendicular  at  the  end 
of  a  line  AB  without  producing  the  line  (Fig. 

Mark  any  point  C  not  in  the  line. 

With  centre  C  and  radius  GB 
describe  an  arc  greater  than  a  semi- 
circumference,  cutting  AB  at  D. 

Join  DC,  and  produce  DC  to 
meet  the  arc  at  E. 

Join  EB ;  EB  is  the  perpendic- 
ular required. 

8.  Draw  a  line  AB  =  4  inches  ;  FlG.  42. 
erect  a  perpendicular  at  each  end, 

making  each  perpendicular,  as  nearly  as  you  can  judge  by 
the  eye,  equal  to  AB.     Test  the  result  with  dividers. 

9.  Draw  a  line  AB ;   erect  at  A  a  perpendicular  equal  to 
4  AB ',  at  B  a  perpendicular  equal  to  ^  AB. 

Work  out  the  exercise  (1)  freehand  ;   (2)  with  instruments. 

10.  Draw  a  horizontal  line  AB  four  inches  long  ;  at  A  erect 
a  perpendicular  AC  =  1^  in.;    at  B  erect  a  perpendicular 
BD  =  4J-  in.     Join  CD ;  measure  CD,  and  write  its  length 
along  its  side. 

Work  the  exercise  (1)  freehand;   (2)  with  instruments. 

11.  Draw  AB  =  5  in.,  J3CJ_ to  AB  and  =  3  in.,  CD  ||  to 
AB  and  =  1  in.     Join  AD,  and  measure  its  length. 

Work  (1)  freehand;   (2)  with  instruments. 

12.  Draw  AB=  6  in.     Upon  AB&$  diameter,  describe  a 
semicircle.     At  a  point  C,  two  inches  from  A,  draw  CD  A.  to 
AB,  and  meeting  the  curve  at  D.    Measure  CD,  and  multiply 
its  length  by  itself.     Record  the  result  thus  :  CD  x  CD= 

If  your  work  were  perfectly  done,  the  product  would  be  8. 
The  nearer  your  result  is  to  8,  the  better  your  work. 

13.  Erect  a  perpendicular  freehand,  and  test  the  result 
with  ruler  and  triangle. 


-° 


\ 


42  LESSONS   IN  GEOMETRY. 


Lesson   2O. 

1.  Problem.  —  To  drop  a  perpendicular  from  a  given 
point  C  to  a  given  line  AB. 

METHOD  I.  (Fig.  43). — Instruments  :  ruler  and  compasses. 

With  centre  C  and  radius  greater 
than  the  distance  from  C  to  AB, 
cut  AB  at  D  and  E.    With  cen- 
G  _        x  ,         tres  D  and  E  and  the  same 
.,-•''  E  B    radius  as  before,  describe  arcs 
,.-•  intersecting  at  F  on  the  other 

,/'  side  of  AB.     Join  CF;    CF  is 

#  -L  to  AB. 

METHOD   II.  —  Instruments  : 

ruler   and  triangle.      This   method   will   be   obvious    from 
Fig.  41. 

2.  Drop  by  Method  I.  a  perpendicular  from  a  point  C  to 
a  line  AB. 

3.  The  same  exercise,  using  Method  II. 

4.  Draw  a  four-sided  figure  ABCD,  and  drop  perpendic- 
ulars from  C  and  D  to  the  side  AB. 

5.  Draw   a  three-sided   figure   and    drop    perpendiculars 
from  the  corners  to  the  opposite  sides  (produced  when  nec- 
essary).    If  your  work  is  accurate,  the  three  perpendiculars 
will  all  pass  through  the  same  point. 

6.  Draw  an  angle  BAC.     Upon  AC  take  AM  =  11  in.  ; 
AN=  2^  in.  ;  AO  =  3|  in.  ;  AP=  5  in.     Drop  perpendicu- 
lars ME,  NS,  OT,  PVto  AB.     Measure  their  lengths,  and 
also  AR,  AS,  AT,  AV.      Record  your  results  in  a  table. 

7.  A  path  goes  straight  for  15  yards,  then  turns  through  a 
right  angle  and  goes  straight  for  40  yards,  then  turns  and 
runs  straight  to  its  starting-point.     Find  its  entire  length  by 
drawing  to  scale  (1  in.  =  10  yds.). 


ANGLES. 


43 


Lesson  21. 

1.  Define  equal  angles. 

Two  angles  are  equal  if  they  can  be  placed  so  that  their 
vertices  coincide  in  position,  and  their  sides  coincide  in 
direction.  For  example,  the  angles  AOB,  BOC,  COD,  etc. 
(Fig.  44),  are  equal  angles. 

2.  What  consequences  follow  from  this  definition  ? 

(1)  All  straight  angles  are  equal;  for  the  sides  of  a  straight 
angle  form  a  straight  line,  and  two  straight  lines  can  always 
be  so  placed  that  they  coincide. 

(2)  All  right  angles  are  equal;   for  a  right  angle  is  half 
of  a  straight  angle,  and  the  halves  of  equal  things  are  also 
equal  (see  Axiom  VIII.). 

(3)  Angles,  like  lines,  can  be  added,  subtracted,  etc. 
For  example  :  in  Fig.  44, 

AOC=AOB  +  BOC 
and    AOB  =  AOC-BOC. 
Also,  the  angles  being  equal, 

AOC=2AOB, 

AOD=3AOB,  etc. 
And  it  is  also  clear  that 


FIG.  44. 


•D 


=  %AOD,  etc. 


3.  Describe  the  units  used  for  measuring  angles. 
The   right   angle   is   divided   into  90   equal  parts   called 

degrees,  the  degree  into  60  equal  parts  called  minutes,  the 
minute  into  60  equal  parts  called  seconds.  Abbreviations  :  the 
sign  °  stands  for  "degree,"  '  for  "minute,"  "  for  "second." 

4.  Reduce  to  degrees  4  rt.  A,  \  rt.  Z,  \  rt.  Z,  \  rt.  Z. 

5.  Reduce  48°  54'  36"  to  seconds  ;   120,000"  to  degrees. 


LESSONS   IN   GEOMETRY. 


6.  Define  equal  arcs. 

Two  arcs  are  equal  if  they  can  be  placed,  one  upon  the 
other,  so  that  they  coincide.  Example  :  the  arcs  AB  and 
CD  (Fig.  45). 

7.  Suppose  a  circumference  (Fig.  4$)  to  be  divided 
into  any  number  of  equal  arcs  by  drawing  radii,  and 
then  compare  the  corresponding  angles  at  the  centre. 

Compare  any  two  adjacent  an- 
gles, as  AOB  and  BOG. 

Imagine  AOB  folded  upon  the 
line  OB  till  A  falls  on  C. 
Why  must  A  fall  on  O? 
Then  OA  will  coincide  with  OC. 
Why  must  OA  and  OC  coincide  ? 
Therefore,  Z  AOB  =  Z  BOG. 
In  the  same  way  all  the  angles 
at  the  centre  can  be  proved  equal. 
Therefore,  if  a  circumference  be  divided  by  radii  into  any 
number  of  equal  arcs,  the  corresponding  angles  at  the  centre 
are  also  equal. 

8.  How  is  this  relation  between  arcs  and  the  corre- 
sponding angles  at  the  centre  applied  in  order  to 
measure  angles? 

The  circumferences  of  circles  are  supposed  to  be  divided 
into  degrees,  minutes,  and  seconds  precisely  in  the  same 
way  as  angles.  A  circumference,  therefore,  contains  360 
degrees  of  arc,  each  degree  contains  60  minutes,  and  each 
minute  contains  60  seconds.  It  follows  that  the  arc,  described 
from  the  vertex  of  an  angle  as  centre,  and  with  any  radius, 
will  contain  the  same  number  of  degrees,  etc.,  as  the  angle. 
And  the  angle  is  measured  by  finding  how  many  degrees, 
etc.,  there  are  in  this  arc. 


FIG.  45. 


ANGLES. 


45 


9,  Why  may  the  arc  employed  to  measure  an  angle 
be  described  with  any  radius? 

It  is  true  that  the  greater  the  radius  the  greater  will  be  the 
arc.  Thus  :  the  arc  AB  (Fig.  46)  is  larger  than  the  arc  CD  ; 
but  if  the  arc  AB  be  divided  into  degrees,  etc.,  and  radii  are 
drawn  to  all  the  points  of  division,  these  radii  will  divide  the 
arc  CD  into  exactly  as  many  equal  parts  as  there  are  in  AB, 
only  the  parts  will  be  smaller.  Therefore,  we  shall  get  the 
same  measure  for  the  angle  whichever  arc  we  use. 


FIG.  47. 

10.  How  are  angles  drawn  on  paper  measured? 
Angles  on  paper  are  measured  with  an  instrument  called  a 

protractor  (Fig.  47).     It  is  a  semicircle,  the  circular  edge  of 
which  is  divided  into  180  degrees. 

To  measure  an  angle,  place  the  centre  of  the  protractor 
over  the  vertex  and  the  zero  line  on  one  side  of  the  angle ; 
then  read  on  the  divided  edge  the  division  through  which 
the  other  side  of  the  angle  passes. 

11.  Make  an  angle  BAG,  and  measure  it  with  the  pro- 
tractor. 

12.  Measure  with  your  protractor  the  angle  given  you. 

13.  Make  with  your  protractor  the  angles  :  30°,  60°,  75°, 
120°,  150°. 

14.  Make  with  your  protractor  the  angles  :  45°,  63°,  107°. 


46 


LESSONS    IN   GEOMETRY. 


Lesson  22. 

1,  Problem.  —  At  a  point  ^C  in  a  line  AB  to  con- 
struct an  angle  equal  to  a  given  angle  DEF' (Fig.  J±8).. 

METHOD  I.  —  Instruments  :  ruler  and  compasses. 

With  centre  E  and  any  radius  describe  the  arc  GH. 

With  centre  C  and  the  same  radius  describe  an  arc  LM, 
cutting  AB  at  L. 

With  centre  L  and  radius  GH,  cut  the  arc  LN  at  N. 

Join  CN-,  Z.  BCN=  Z  DEF. 

METHOD  II.  —  Instruments  :  ruler  and  protractor. 
This  method  is  obvious  without  explanation. 

Ji/F 


AC  L       B 

FIG.  48. 


FIG.  49. 


2.  Problem.  —  To  bisect  a  given  angle  BAC  (Fig. 
With  centre  A  describe  the  arc  DE. 

With  centres  D  and  E,  and  the  same  radius  as  before, 
describe  arcs  intersecting  at  F. 

Draw  AF-,  the  line  AF  bisects  the  Z.  BAC. 

3.  Make  an  angle  BAC,  and  then  construct  with  ruler  and 
compasses  an  angle  DEF  equal  to  BAC. 

4.  Make  with  your  protractor  an  angle  of  40° ;  then  con- 
struct an  equal  angle  with  ruler  and  compasses. 

5.  Construct  a  right  angle,  and  then  bisect  it.     What  is 
the  value  of  each  of  the  equal  parts  ? 


ANGLES. 


47 


6.  Problem.  —  To  construct  an  angle  of  60°  (Fig.  50). 

Draw  a  line  AB.  E 

With  centre  A  and  any  radius 
describe  the  arc  CD. 

With  centre  (7,  and  the  same  radius, 
cut  the  arc  CD  in  E. 

Join  AE ;  Z  BAE  =  60°. 


C 

FIG.  50. 


7.  Construct  an  angle  of  60°,  and  then  bisect  it. 

8.  Construct  an  angle  BAG  =  60°  ;   from  C  drop  CD  _L 
to  ^4J5  ;  and  find  the  value  of  the  angle  ACD. 

9.  Construct  an  angle  BAC=  60°  ;  join  BC ;  and  find  the 
values  of  the  angles  ABC  and  ACB. 

10.  Construct  with  ruler  and  compasses  an  angle  of  30°. 
Verify  your  result  with  your  protractor. 

11.  Construct  with  ruler  and  compasses  an  angle  of  221°. 
Verify  with  your  protractor. 

12.  Draw  any  acute  angle  BAC ;  then  construct  an  angle 
three  times  as  large. 

13.  Draw  freehand  the  angles  30°,  45°,  GO0.    Correct  your 
results  with  your  protractor.     Record  your  errors  in  degrees. 

14.  Make  an  angle,  and  bisect  it  freehand.    Correct  your 
result  with  your  protractor.     Record  your  error. 

15.  Draw  a  three-sided  figure  ABC.     Find  the  values  of 
its  angles,  (1)  by  estimating  them  by  the  eye,  (2)  by  meas- 
uring  them  with   your  protractor.     Record   your  results  in 
tabular  form,  thus  :  — 


BAC 

ABC 

ACB 

Sum. 

Estimated  Value. 
Measured  Value. 

Difference. 

48  LESSONS   IN   GEOMETRY. 

Lesson  23. 

1.  Define  complementary  and  supplemental^  angles. 
Two  angles  are  said  to  be  complementary  if  their  sum  is 

equal  to  a  right  angle,  or  90°,  and  each  is  called  the  comple- 
ment of  the  other.     Example  :  67°  and  23°. 

Two  angles  are  said  to  be  supplementary  if  their  sum  is 
equal  to  a  straight  angle,  or  180°,  and  each  is  called  the 
supplement  of  the  other.  Example  :  150°  and  30°. 

2.  Define  adjacent  angles.     When  are  they  supple- 
mentary ? 

Two  angles  which  have  the  same  vertex  and  a  common 
side  between  them  are  called  adjacent  angles. 

If  a  line  OO  (Fig.  51)  is  drawn  from  any  point  0  in  AB, 
two  adjacent  angles  AOC,  BOO  are  formed ;  and  whatever 
be  the  direction  of  OC  it  is  obvious  that 
AOC+COB=180°. 

The  two  adjacent  angles  formed  when  one  straight  line 
meets  another  are  supplementary. 


BOA 

FIG.  51.  FIG.  52. 

3.  What  is  the   complement  of  35°?  60°?  1°?  89°?  45°? 

4.  What  is  the  supplement  of  60°?  90°?  1°?  135°?  62°  12'? 

5.  Draw  two  angles  adjacent  but  not  supplementary. 

6.  If  OD  (Fig.  51)  is  _L  to  AB,  what  is  the  complement 
of  Z  .400? 

7.  If  Z  .400=  48°,  find  Z  COD  and  Z  COB. 


ANGLES.  49 

8.  Define  vertical  angles. 

Two  angles  are  called  vertical  angles  if  one  of  them  is 
formed  by  producing  the  sides  of  the  other  from  the  vertex. 

When  two  straight  lines  intersect  (Fig.  52)  two  pairs  of 
vertical  angles,  a  and  c,  b  and  c?,  are  formed. 

9.  Two  vertical  angles  are  equal.    How  can  this  be 
shown  to  be  true? 

We  may  use  three  different  methods  : 

(1)  Measure  them  directly  with  a  protractor. 

(2)  Cut  them  out,  and  place  one  upon  the  other. 

(3)  Observe  that  the  vertical  angles  a  and  c  (Fig.  52) 
both  have  the  angle  b  for  their  supplement,  and  then  reason 
as  follows : 

Since  a=180°-&, 

and  c  =  180°  -  6, 

therefore  a  =  c.  (Axiom  1). 

10.  What  advantages  has  the  third  method? 

The  third  method  requires  no  instruments ;  it  employs 
only  reasoning.  Moreover,  it  is  easy  to  see  that  this  reason- 
ing applies  equally  well  to  the  vertical  angles  b  and  d,  or  to 
any  pair  of  vertical  angles  whatever.  Therefore  it  convinces 
us  immediately  that  vertical  angles  are  always  equal. 

11.  If  Z  a  =  37°  (Fig.  52),  find  the  A  b,  c,  and  d. 

12.  Two  lines  intersect  so  that  one  of  the  angles  formed 
is  90°.     What  are  the  values  of  the  other  angles? 

13.  Draw  any  two  intersecting  lines,  and  find  the  values 
of  the  four  angles  which  are  formed. 

14.  Write  out  the  proof  by  the  third  method  employed  in 
No.  9  that  Z  b  =  Z  d  (Fig.  52) . 


50  LESSONS   IN   GEOMETRY. 


Lesson  24. 

1.  Name  certain   pairs  of  angles  formed  when  a 
straight  line  EF  cuts  two  parallel  lines  AB  and  CD 
(Fig.  53). 

The  line  EF  forms  with  AB  four  angles,  a,  b,  c,  d,  and 
with  CD  four  angles,  m,  n,  o,  p. 

The  pair  of  angles,  a  and  m,  and  the  other  pairs  similarly 
placed  with  respect  to  AB  and  CD,  are  called  exterior-interior 
angles.  The  pairs,  c  and  m,  d  and  n,  are  called  alternate- 
interior  angles. 

2.  Draw  two  parallels.     Cut  them  by  a  third  line.     Mark 
the  angles  about  the  points  of  intersection  1,  2,  3,  4  and  5, 
6,  7,  8.     Name  the  exterior-interior  angles  and  the  alternate- 
interior  angles. 

E 


A  y^         B 

n/m . 

C  <J/P  D 


B 

m 


D 


F 
FIG.  53.  Fio.  54. 

3.  The  exterior-interior  angles  are  equal.  How  can 
this  be  shown  without  use  of  instruments  ? 

If  (Fig.  53)  AB  be  moved  towards  CD,  keeping  all  the 
time  parallel  to  AB,  the  angles  m,  w,  o,  p  will  not  change, 
because  their  sides  do  not  change  in  direction.  Let  the 
motion  continue  till  AB  and  CD  cut  EF  at  the  same  point ; 
then  AB  and  CD  will  coincide  (see  Axiom  10  for  the  reason) . 
Therefore  a  will  coincide  with  m,  b  with  n,  c  with  o,  d  with  p. 
In  other  words,  a  =  m,  6  =  n,  c  =o,  c?  =  p. 


ANGLES.  51 

4.  The  alternate-interior  angles  are  equal.   How  can 
this  be  shown  to  be  true? 

Compare  the  alternate-interior  angles  c  and  m  (Fig.  53). 

We  have  jnst  shown  that     m  =  a. 

We  know  also  that  c=  a  (p.  49,  No.  9). 

Therefore,  by  Axiom  1,         c  =  m. 

In  the  same  way  we  can  show  that  d  =  n. 

5.  If  two  straight  lines  AB,  CD,  cut  a  third  line  EF 
so  that  the  exterior-interior  angles,  or  the  alternate- 
interior  angles  are  equal,  the  two  lines  are  parallel. 
How  can  this  be  shown  to  be  true? 

(1)  Suppose  the  exterior-interior  angles,  a  and  m,  equal 
(Fig.  53).     If  AB  be  moved  towards  CD  without  change  of 
direction  till  both  lines  cut  EF  at  the  same  point,  the  two 
lines  will   then  coincide,  since   they  will   pass  through  the 
same  point  and  make  equal  angles,  a  and  m,  with  the  same 
portion  of  the  line  EF. 

Therefore  AB  and  CD  must  have  the  same  direction. 
Therefore  AB  and  CD  are  parallel  (p.  18,  No.  1). 

(2)  Suppose  the  alternate-interior  angles,  c  and  m,  equal. 
The  angles  c  and  a  are  equal.     Why?     Therefore  the  angles 
a  and  m  are  equal,  and  the  reasoning  in  (1)  holds  good. 

6.  If   (Fig.  53)    a  =  40°,    find    the   values   of   the    other 
angles. 

7.  If  (Fig.  54)  AB  is  II  to  CD,  and  EF  is  _L  to  AB,  is 
EF  also  J_  to  CD  ?     What  reason  can  you  give  ? 

8.  If  (Fig.  54)   AB  and  CD  are  each  _L  to  EF,  does  it 
follow  that  AB  is  II  to  OD?     Why? 

9.  Draw  two  parallels.     Cut  them  by  a  third  line.     Then 
find  the  values  of   all  the   angles  which  are  formed,  using 
your  protractor  as  little  as  possible. 


52  LESSONS   IN   GEOMETRY. 

Lesson  25.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  18-20. 

2.  Can  two  horizontal  lines  be  _L  to  each  other? 

3.  A  horse  running  clue  northeast  turns  to  the  left  through 
li  right  angles.     In  what  direction  is  he  now  running? 

4.  Can  you  find  a  way  to  test  whether  the  edges  of  your 
triangle  are  exactly  J_  to  each  other  ? 

5.  Draw  a  line,  and  erect  a  _L  by  folding  the  paper. 

6.  Erect  a  _L  freehand.     Find,  with  protractor,  your  error. 

7.  Draw  a  three-sided  figure,  and  erect  perpendiculars  at 
the  middle  points  of  the  sides. 

8.  Describe  a  circle,  draw  a  diameter,  and  join  any  three 
points  in  the  circumference  to  the  ends  of  the   diameter. 
Measure    the    three    angles    formed    at    the    three    points. 
Do  your  results  seem  to  indicate  any  general  truth? 

9.  A  garden  has  the  shape  of  the  annexed  figure.     Its 

„  _  dimensions  are  as  follows  : 

^ 

Length,  AB=300  ft. 
Breadth,  5(7=200  ft. 


Draw  a  plan  to  the  scale  1  : 1,200.     Join 
— g  AC,  BD,  and  find  their  lengths. 


FIG.  55.  Do  AG  and  BD  bigect  each  other  ? 


10.  A  man  walks  2  miles  ;  then  turns  to  his  right,  through 
a  right  angle,  and  walks  3  miles  ;    then   turns  to  his  left, 
through  a  right  angle,  and  walks  a  mile.     Draw  a  plan,  and 
find  his  distance  from  his  starting-point  (scale,  1  in.  =  l  mile). 

11.  A  path  goes  straight  for  3  yds.  ;    then  goes  at  right 
angles  for  8  yds.  ;    then  goes  straight  to  the  middle  point 
between  where  it  started  and  where  it  first  turned  off.     Find 
its  length  by  drawing  to  scale. 


ANGLES.  53 

Lesson  26.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  21-24. 

2.  Add  together  28°  39',  37°  48'  35",  and  78°  9'  55". 

3.  Find  the  difference  between  the  complements  of  28°  5' 
and  37°  27'. 

4.  What  is  the  supplement  of  five  times  17°  21'? 

5.  If  seven  equal  angles  are  constructed  about  a  point, 
find  the  value  of  each  angle  correct  to  the  nearest  minute. 

6.  Describe  a  circle  with  centre  0.     Then  construct  with 
protractor  the  angles  AOB=  15°,  BOG  =  37°,   COD  =  50°, 
DOE  =  90°,  EOF=  110°.     Find  the  value  of  Z.FOA  (1)  by 
the  eye  ;   (2)  by  the  protractor  ;  (3)  by  calculation. 

7.  Construct  with  ruler  and  compasses  an  angle  of  120°. 

8.  Make  two  supplementary  adjacent  angles,  and  bisect 
each  of  them.     Then  measure  the  angle  formed  by  the  two 
bisectors. 

9.  Trisect  a  right  angle  (see  p.  47,  No.  6). 

10.  Make  any  angle   BAC,  a  point  P,  and  a  line  DE. 
Then  draw  through  P  a  line  making  with  DE  an  angle  equal 
to  BAC.     (Apply  No.  1,  p.  46  ;  No.  4,  p.  19  ;  No.  3,  p.  50.) 

11.  Construct  an  angle  BAC  =  30°.     From  any  point  D 
in  AC  drop  DE  _L  to  AC  and  meeting  AC  at  E.     Measure 
AD  and  DE. 

12.  A  man  walks  2  miles.     He  then  turns  to  the   right 
through  an  angle  of  60°,  and  walks  2  miles.     Again  he  turns 
to  the  right  through  60°,  and  walks  2  miles.     How  far  is  he 
now  from  his  starting-point?     Construct  to  scale. 

13.  A  man  walks  6  miles,  turns  through  45°  to  his  left, 
walks  1  mile,  turns  through  90°  to  his  left,   and  walks  2 
miles  further.     How  far  is  he  now  from  home  ?     Construct 
to  scale,  and  find  the  answer  correct  to  eighths  of  a  mile. 


54  LESSONS   IN   GEOMETRY. 

Lesson  27.    Keview. 

1.  Review  all  the  italicized  exercises  in  Chapter  III. 

2.  Draw  AB  =  50mm;  erect  at  a  point  C,  lcm"from  B,  a 
perpendicular  <7Z>  =  40mm;    join  DB  and  DA,  and  find  in 
millimeters  the  difference  between  DB  and  DA. 

3.  Describe  a  circle  with    centre    0   and   radius  =  60mm. 
Draw  any  radius  OA,  bisect  it  at  C,  and  let  D,  E  be  the 
points  where  the  line  of  bisection  meets  the  circumference. 
Join  OD,  OE.     Measure  the  length  DE,  and  also  measure 
with  the  protractor  the  angles  ODE,  OED,  DOE. 

4.  Draw  AB  =  80mm.     Bisect  it  at  G.     Erect  perpendic- 
ulars at  A,  B,  C.     Draw  an}'  line  cutting  these  perpendicu- 
lars  at   D,    E,   F,   respectively.     Measure   AD,  BE,   CF. 
Find  with  protractor  the  sum  of  the  angles  ADE  and  BED. 

5.  Draw  any  line  AB.     Erect  the  bisecting  perpendicular. 
Join  any  two  points  D,  E  in  this  perpendicular  to  A  and  B. 
Measure   AD,  BD,  AE,  BE.     Is   any  general   truth   sug- 
gested by  your  results? 

6.  Three  ships  start  from  the  same  port,  sailing  N.,  E.,  S., 
at  the  rates  6,  8,  10  kilometers  an  hour,  respectively.     How 
far  are  they  from  one  another  at  the  end  of  4  hours  ?     Find 
the  distances  correct  to  half -kilometers  (scale,  2mm  =  lkm). 

7.  Two   cities    are    connected    by   a   road   which    forms 

a  broken  line  ABODE 
(Fig.  56).  AB=  5km, 
BC.=  3.75km,  OD=3kin, 
DE  =  4km  ;  Z  ABC  = 
90°,  Z  BCD  =  90°,  Z 
ODE  =  135°.  A  new 
road  is  made,  going- 
straight  from  A  to  C.  Find  by  drawing  to  scale  the  dis- 
tance from  A  to  E  by  the  new  road. 


CHAPTER   IV. 


TRIANGLES. 


Lesson  28. 

1.  Define  a  triangle,  Us  sides,  perimeter,  vertices. 
A  triangle  is  a  plane  figure  bounded  by  three  straight 

lines.  The  straight  lines  are  called  its  sides;  their  sum,  its 
perimeter ;  their  points  of  intersection,  its  vertices. 

The  sides  form  three  angles ;  the  three  sides  and  the 
three  angles  are  called  the  parts  of  the  triangle. 

The  sign  A  stands  for  "  triangle,"  A  for  "  triangles." 

2,  How  are  triangles  divided  with  respect  to  their 
sides  ? 

There  are  three  classes  ;  namely  : 
Equilateral  triangles,  having  all  the  sides  equal. 
Isosceles  triangles,  having  two  sides  equal. 
Scalene  triangles,  having  no  two  sides  equal. 


EQUILATERAL.  ISOSCELES.  SCALENE. 

3.  Draw,  freehand,  a  triangle,   and  name  its  six   parts, 
using  only  the  letters  placed  at  the  vertices. 

4.  Draw,  freehand,  two  equilateral  triangles,  two  isosceles 
triangles,  and  two  scalene  triangles. 


56 


LESSONS  IN   GEOMETRY. 


5,   How  are  triangles  divided  with  respect  to  their 

angles  ? 

As  regards  angles  there  are  three  classes  ;  namely  : 

Acute  triangles,  having  all  the  angles  acute. 

Obtuse  triangles,  having  one  obtuse  angle. 

Right  triangles,  having  one  right  angle. 

In  a  right  triangle  the  side  opposite   the  right  angle  is 
called  the  hypotenuse,  and  the  other  sides  are  called  the  legs. 


ACUTE. 


OBTUSE. 


RIGHT. 


6.  Draw,  freehand,  two  acute  triangles,   two   obtuse  tri- 
angles, and  two  right  triangles. 

7.  How  are  the  dimensions  of  a  triangle  named  ? 
The  two  dimensions  are  named  base  and  altitude. 

In  the  isosceles  triangle  the  side  not  equal  to  the  others  is 
called  the  base,  and  the  equal  sides  are  called  the  legs. 

In  other  kinds  of  triangles  any  side  may  be  taken  as  base. 

In  all  triangles  the  alti- 
tude is  the  perpendicular 
dropped  to  the  base  from 
the  opposite  vertex. 

In  Fig.  57  the  three 
altitudes  are  all  drawn : 
notice  that  they  meet  in 
one  point. 


N 


B      M 

FIG.  57. 

In  an  obtuse  triangle  two  of  the  altitudes  lie  oulside  the 
figure,  and  meet,  not  the  base,  but  the  base  produced. 

Thus,  in  the  obtuse  triangle  MNO  (Fig.  57)  the  altitude 
OP  meets  the  base  MN  produced  at  P. 


TRIANGLES.  57 

8.  Draw  an  acute  triangle,  and  construct  the  altitudes. 

9.  Draw  an  obtuse  triangle,  and  construct  the  altitudes. 

10.  Draw  a  right  triangle,  take  the  hypotenuse  as  base, 
and  construct  the  altitude.     What  is  the  altitude  if  one  of 
the  legs  is  taken  as  the  base? 

11.  Draw  any  two  lines  a  and  6;    then  make  a  triangle 
having  a  base  equal  to  a  and  an  altitude  equal  to  b.     Can 
you  make  more  than  one  such  triangle? 

12.  The  perimeter  of  an  isoceles  A  =17  in.     Find  (1) 
the  base  if  one  leg  =  5  in.  ;   (2)  each  leg  if  the  base  =  5  in. 

13.  Draw  a  A  ABC.     Which  is  the  greater,  AB  or  AC+ 
CB?     Why?     BCorAB+AC?     ACorAB  +  BC?    What 
is  true  o'f  the  sum  of  two  sides  of  any  triangle  ? 

14.  Problem.  —  To  construct  a  right  triangle  having 
given  the  lengths  of  the  hypotenuse  and  one  leg. 

Let  a  and  b  be  the  given 
lengths. 

Draw  AB  equal  to  b. 

At  B  erect  a  _L  ED. 

With  centre  A  and  radius  a 
cut  BD  at  C. 

*A.  jj 

Join  AC  ;  then  the  A  ABC  is  FlG  58 

the  A  required. 

15.  Construct  a  right  triangle  having  given  the  hypotenuse 
3f  in.,  one  leg  2J  in.     Measure  the  other  leg. 

16.  Construct  right  triangles  having  for  legs  (1)  3  in.  and 
4  in.  ;    (2)    11  in.   and    2  in.      Measure   the   hypotenuses. 
Compare  your  results. 

17.  On  the  ground  in  front  of  a  wall  is  a  flower-bed  15  ft. 
wide.     It  requires  a  ladder  25  ft.  long  to  reach  from  the 
outer  edge  of  the  bed  to  the  top  of  the  wall.     How  high  is 
the  wall?     Solve  by  drawing  to  scale. 


58 


LESSONS   IN   GEOMETRY. 


IJesson  29. 

1.  Problem.  —  To  construct  a  triangle,  having  given 
the  lengths  of  the  three  sides  (Fig.  59). 

Let  a,  6,  c  be  the  lengths. 

Draw  AB  equal  to  a. 

"With  centre  A  and  radius  b 
describe  an  arc  DC. 

With  centre  B  and  radius  c 
describe  an  arc  EC,  cutting  the 
arc  DC  atC.  Join  AC  and  BC. 
A  ABC  is  the  A  required. 


B 


FIG.  59. 


2.  Construct  a  triangle,  with  sides  4  in.,  3^  in.,  2-|-  in. 

3.  Construct  a  triangle,  with  sides  4  in.,  3  in.,  3  in. 

4.  Construct  an  isosceles  triangle,  given  :    base  =  3  in.  ; 
one  leg  =  5  in. 

5.  Construct  an  equilateral  triangle,  having  given  one  side. 

6.  Construct  an  equilateral  triangle  with  a  perimeter  equal 
to  8  in. 


7.  Problem.  —  To  construct  a  triangle,  having  given 
two  sides  and  the  included  angle  (Fig.  60). 

Let  a,  6,  be  the  sides,  and  Z  m  the  given  angle  (Fig.  60) . 

a Draw  a  straight  line,  and  at 

any  point  A  of  the  line  con- 
struct an  angle  equal  to  m. 

Upon  its  sides  lay  off  AB  = 
a,  AC=b.  Join  BC.  A  ABC 
is  the  A  required. 


B 


FIG.  60. 


NOTE.  —  In  Fig.  60  the  auxiliary  lipes  used  for  constructing  the  angle 
are  not  given.     Draw  auxiliary  lines  dotted  and  very  fine. 


TRIANGLES.  59 


8.  Construct  a  triangle,  given  AB  =  3  in.,  AC  =  3^  in., 
Z  A  =  45°. 

9.  Construct  an  isosceles  right  triangle,  having   given 
one  leg. 

10.  Explain  how  the  distance  between  two  points, 
situated  on  opposite  sides  of  a  pond,  can  be  measured. 

Let  A  and  B  (Fig.  61)  be  the  points. 

First  choose  a  point  C  from  which  A  and  B  are  both  visible  ; 
then  measure  AC,  BC,  and  Z  ACB  ;  then  construct  to  scale 
on  paper  a  triangle  acb,  mak- 
ing Z  ac&  =  Z  ACB,  and  taking 
ac  of  the  proper  length  to  rep- 
resent AC,  and  be  to  repre- 
sent BC.  Measure  ab,  and 
multiply  it  by  the  reducing 
factor  you  have  used.  The 
product  will  be  the  distance 
from  A  to  B.  FlG.  61. 

Instead   of  constructing   to 

scale  on  paper,  if  the  ground  is  level  and  clear,  we  may 
produce  AC  to  D,  making  CD  =  CA,  and  BC  to  E,  making 
CE  =  CB.  Then  measure  ED.  The  distance  from  E  to  D 
will  be  the  same  as  the  distance  from  A  to  B. 

11.  Suppose   your  notes  of  measurement  made   to  find 
AB  (Fig.  61)    to  be:    AC  =1200  yds.,    £(7=1500  yds., 
Z  ACB=  110°.     Find  the  distance  from  A  to  B. 

12.  Two  villages  A  and  B  are  separated  by  a  river.    There 
is  a  straight  road  going  from  each  village  to  a  bridge  over 
the  river  at  C.     If  AC=  5  miles,  BC=  6  miles,  Z  ACB  = 
30°,  find  how  far  A  is  from  B. 

13.  Newton  is  7  miles  west  of  Boston  ;  Arlington  is  6  miles 
northwest  of  Boston.     How  far  is  Newton  from  Arlington? 


60 


LESSONS   IN   GEOMETRY. 


Lesson  3O. 

1.  Problem.  —  To  construct  a  triangle,  having. given 
one  side  and  the  two  adjacent  angles  (Fig.  62). 

Let  a  be  the  given  side  ;  w 
and  n  the  given  angles. 
Draw  AB  equal  to  a. 
At  A  construct  /.  BAC=m. 
At  B  construct  Z.  ABC—n. 
Produce   the  lines  AC  and 
FIG.  62.  BC  till  they  meet  at  C. 

A  ABC  is  the  A  required. 

2.  Construct  a  triangle,  given   AB=3  in.,  BAC=±0°, 
ABC  =80°. 

3.  Define  angle  of  elevation  and  angle  of  depression. 
If  a  line  is  drawn  from  a  point  to  the  eye  of  an  observer, 

the  angle  which  this  line  makes  with  the  horizontal  plane 
passing  through  the  eye  is  the  angle  of  elevation  or  the  angle 
of  depression  of  the  point,  according  as  the  point  is  above  or 
below  the  horizontal  plane. 

For  example  (Fig.  63)  :  if  the  eye  is  at  (7,  the  angle  of 
elevation  of  A  is  the  angle  ACB ;  if  the  e}Te  is  at  A,  the 
angle  of  depression  of  C  is  the  angle  DAG. 

D  A 


B 


4.  What  is  the  angle  of  depression  of  E  as  seen  from  A? 

5.  What  pairs  of  angles  in  Fig.  63  are  equal?    Why? 


TRIANGLES.  61 

6.  Construct  a   right  A,   given   one   leg  =  2  J  in.,   one 
angle  =  50°. 

7.  Construct 'a  right  A,  given  hypotenuse  =  5  in. ,  one 
angle  =  50°. 

8.  Construct  an  isoceles  A,  given  the  base,  and  angle 
at  the  base. 

9.  Find  the  height  of  a  church  spire,  if  the  angle  of  ele- 
vation of  the  top,  100  yds.  from  the  bottom,  is  45°. 

10.  At  a  distance  of  150  ft.  from  the  foot  of  a  tree,  I 
find  the  angle  of  elevation  of  the  top  to  be- 30°.     How  high 
is  the  tree  ? 

11.  Wishing  to  find  the  height  AB  of  a  tower  (Fig.  63), 
I  observe  the  angle  of  elevation  of  the  top  at  a  point  E,  and 
find  it  to  be  30°.     At  a  point  F,  120  ft.  nearer  the  tower,  I 
find  that  the  angle  of  elevation  of  A  is  45°.     What  is  the 
height  of  the  tower? 

12.  Wishing  to  find  the  breadth  AB  of  a  river,  I  measure 
on  a  line  _L  to  AB  along  the  bank  a  distance  of  400  ft.,  and 
find  that  Z  ACB  =  70°.     What  is  the  breadth  of  the  river  ? 

13.  A  lighthouse  A  stands  on  an  island.     To  find  its  dis- 
tance from  the  shore  a  man  runs  a  base  line  BC  equal  to 
1000  yds.  along  the  shore,  and  measures  the  angles  ABC, 
ACB.     He  finds  /.  ABC  =80°,    Z  ACB  =70°.     Find  the 
distance  from  B  to  the  lighthouse. 

NOTE.  —  The  learner  who  has  carefully  worked  out  the  exercises  in 
Lesson  29  and  in  the  present  lesson  will  be  able  to  understand  what  is 
meant  by  the  indirect  measurement  of  a  line.  To  measure  any  consid- 
erable distance  by  yard-stick,  chain,  or  other  direct  means,  would  be  a 
very  laborious  process.  In  many  cases,  also,  direct  measurement  is 
practically  impossible  on  account  of  obstacles,  such  as  water,  swamps, 
etc.  In  such  cases,  Geometry  teaches  us  how  to  measure  the  line  indi- 
rectly, by  treating  it  as  a  part  of  a  triangle  which  we  can  construct  after 
we  have  found  the  values  of  three  other  parts  by  direct  measurement. 


62 


LESSONS   IN   GEOMETRY. 


Lesson  31. 

1.  How  can  it  be  shown  to  the  eye  that  the  sum  of 
the  angles  of  a  triangle  is  equal  to  a  straight  angle, 
or  180°? 

Draw  a  triangle  ABC  (Fig.  64)  and  the  altitude  CD.  Cut 
out  the  triangle,  and  fold  the  corners  over  the  dotted  lines 
till  they  come  together  at  D.  The  three  angles  of  the  tri- 
angle will  just  make  a  straight  angle  at  D. 

C  D  C  E 


B 


2.  Prove  that  the  sum  of  the  angles  of  every  tri- 
angle is  equal  to  180°. 

Draw  any  triangle  ABC  (Fig.  65)  ;  let  a,  6,  c,  denote  its 
angles. 

We  wish  to  prove  that  a  +  b  +  c  =  180°. 

Through  C  draw  DE  ||  to  AB.  Let  m  denote  the  angle 
DC  A,  and  n  the  angle  ECB. 

The  A  a  and  m  are  equal,  because  they  are  alt. -int.  angles. 

The  A  b  and  n  are  equal  for  the  same  reason. 

Therefore,  a  +  6  +  c  =  ra  +  n  +  c.  (Ax.  2.) 

But  m  +  n  -f  c  ==  a  straight  angle,  or  180°. 

Therefore,  a  +  b  +  c  =  180°.  (Ax.  1 .) 

This  reasoning  applies  equally  well  to  any  triangle  ;  there- 
fore the  sum  of  the  angles  of  every  triangle  is  equal  to  180°. 


TRIANGLES.  63 

3.  State   three   truths  which  follow  immediately 
from  what  has  just  been  proved. 

(1)  If  two  angles  of  a  triangle  are  equal,  respectively,  to 
two  angles  of  another  triangle,  then  the  third  angles  must  also 
be  equal. 

For  the  third  angle  in  each  case  is  found  by  subtracting 
the  sum  of  the  other  two  from  180° ;  and  the  sum  of  the 
other  two  in  each  case  is  the  same  (Ax.  2)  ;  and  if  equals  be 
taken  from  equals,  the  remainders  will  be  equal  (Ax.  3). 

(2)  A  triangle  can  have  only  one  right  angle  or  one  obtuse 
angle. 

For  two  right  angles  would  make  180°,  and  two  obtuse 
angles  would  make  more  than  180°. 

(3)  The  two  acute  angles  of  a  right  triangle  are  comple- 
mentary;   that  is,  their  sum  is  always  90°. 

For  their  sum  is  found  by  subtracting  the  right  angle,  or 
90°,  from  180°,  and  180°-  90°=  90°. 

4.  If  one  angle  of  a  triangle  is  45°,  what  is  the  sum  of 
the  other  two  angles  ? 

5.  If  two  angles  of  a  triangle  are  37°  and  75°,  find  the 
third  angle. 

6.  If  two  angles  of  a  triangle  are  64°  47'  33"  and  77°  18' 
41",  find  the  third  angle. 

7.  If  one  of  the  acute  angles  of  a  right  triangle  is  30°, 
what  is  the  other?     If  it  is  27°  5',  what  is  the  other? 

8.  If  the  acute  angles  of  a  right  triangle  are  equal,  what 
is  the  value  of  each? 

9.  Find  the  value  of  each  angle  in  an  equiangular  triangle. 
10.    Draw  a  triangle  (the  larger  the  better) ,  measure  its 

angles,  and  find  their  sum.     If  the  sum  is  not  exactly  180°, 
what  is  the  reason  ? 

NOTE  TO  THE  TEACHER.  —  The  pupils  should  work  No.  10  together, 
and  the  results  should  be  put  on  the  board  and  compared  with  180°. 


64 


LESSONS   IN   GEOMETRY. 


Lesson  32. 

1.  Compare  two  triangles  with  respect  to  size  and 
shape. 

There  are  four  cases,  as  follows : 

(1)    Two  triangles  may  have  the  same  size  (magnitude). 
They  may  have  the  same  shape  (form). 
They  may  agree  both  in  size  and  shape. 
(4)    They  may  differ  both  in  size  and  shape. 
For  example  (Fig.  66)  :    the   triangles  ABC  and  DEF 
have  the  same  size  ;  ABC  and  MNO  have  the  same  shape  ; 
ABC  and  PQR  agree  both  in  size  and  shape  ;  and  DEF  and 
MNO  differ  both  in  size  and  shape. 


(2) 
(3) 


.D       „  E      M 

FIG.  66. 

2.  Define  equivalent  triangles;   similar  triangles; 
equal  triangles. 

Two  triangles  are  equivalent,  if  they  have  the  same  size. 
Two  triangles  are  similar,  if  they  have  the  same  shape. 
Two  triangles  are  equal,  if  they  agree  in  size  and  shape. 

3.  To  what  extent   do   these  definitions  apply  to 
other  magnitudes  than  triangles? 

They  apply,  in  general,  to  lines,  to  surfaces,  and  to  solids. 

Thus,  a  curved  line  may  have  the  same  length  as  a  straight 
line  ;  a  field  bounded  by  curved  lines  may  enclose  the  same 
extent  as  a  field  bounded  by  straight  lines  ;  a  cubical  vessel 
may  hold  the  same  quantity  of  water  as  a  vessel  shaped  like 


TRIANGLES.  65 

a  cylinder.  In  all  these  cases,  the  size  is  the  same,  the  shape 
different.  They  are  examples  of  equivalent  magnitudes. 

Again,  two  straight  lines  have  the  same  shape  whatever 
be  their  lengths  ;  so  have  two  circles,  two  cubes,  or  two 
spheres,  however  much  they  differ  in  size.  They  are  similar 
magnitudes. 

If  two  straight  lines  have  the  same  length  they  are  equal 
magnitudes ;  so  also  are  two  circles,  or  two  spheres,  which 
have  the  same  size. 

Of  these  three  kinds  of  agreement,  —  equivalence,  simi- 
larity, and  equality,  —  the  last  is  the  simplest,  and  should 
be  studied  first. 

4.  The  study  of  equal  triangles  raises  two  questions. 
What  is  the  first  question ,  and  what  is  the  answer  ? 

If  we  know  that  two  triangles  are  equal,  what  can  we  infer 
respecting  their  sides  and  their  angles  ? 

The  answer  is  obvious.  Since  two  equal  triangles  agree 
in  size  and  in  shape,  they  can  differ  only  in  position;  and  if 
placed  one  upon  the  other,  they  must  coincide  in  all  their  six 
parts.  In  other  words,  their  sides  and  angles  must  be  equal, 
each  to  each.  In  the  equal  A  ABC  and  PQR  (Fig.  66), 

AB=PQ,  AC=PR,  BC=QR; 

^ACB  =  ZPRQ,    Z.ABC=^PQR,    Z.  BAG '  =  Z  QPR. 

In  any  two  equal  triangles,  the  sides  opposite  tivo  equal 
angles  are  equal;  two  such  sides  are  called  homologous  sides. 

5.  Draw,  freehand,  two  equal  triangles.    Name  the  equal 
angles,  pair  by  pair.     Name  the  equal  or  homologous  sides. 

6.  What  is  the  other  question  about  equal  triangles  ? 
How  many,  and  what  parts,  of  two  triangles  must  be  known 

to  be  equal,  in  order  that  we  may  conclude,  without  error, 
that  the  triangles  themselves  are  equal? 

To  answer  this  question,  special  cases  must  be  examined. 


66  LESSONS    IN   GEOMETKY. 

Make  two  unequal  triangles : 

7.  Each  having  a  given  length  for  one  side. 

8.  Each  having  two  given  lengths  for  two  sides. 

9.  Each  having  a  given  angle. 

10.  Each  having  two  given  angles.     What  is  true  of  the 
remaining  pair  of  angles  ?     Why  ? 

11.  Each  having  a  given  length  for  one  side,  and  also  a 
given  angle  adjacent  to  this  side. 

12.  Each  having  a  given  length  for  one  side,  and  a  given 
angle  opposite  to  this  side.     Make  the  angle  first. 

13.  Is  a  triangle  determined  if  one  part  is  given?     If  two 
parts  are  given  ?     If  the  three  angles  are  given  ? 

14.  Theorem.  —  Two  triangles  are  equal,  if  two  sides 
and  the  included  angle  of  one  are  equal,  respectively, 
to  two  sides  and  the  included  angle  of  the  other, 

HYPOTHESIS. 

A  ABC  and  DEF, 
AB=DE,  AC=DF, 


CONCLUSION.  -^  B  D 

A  ABC  =  A  DEF.  FlQ' 67' 

PROOF.    We  may  divide  the  proof  into  five  steps. 

(1)  Imagine  A  ABC  placed  on  A  DEF  so  that  A  falls 
on  D,  and  the  equal  angles  coincide. 

(2)  Then  E  will  fall  on  B,  and  F  on  C.    Why? 

(3)  Therefore  EF  will  coincide  with  BC.    Why? 

(4)  Therefore  the  two  triangles  will  coincide.    Why  ? 

(5)  Therefore  A  ABC  =  A  DEF. 

Since  this  reasoning  applies  equally  well  to  any  two  tri- 
angles, the  theorem  is  proved. 

15.    Name  all  the  equal  parts  in  the  A  ABC  and  DEF. 


TRIANGLES.  67 

Lesson  33. 

1,  What  is  a  theorem,  and  what  are  its  parts? 

A  theorem  is  a  statement  to  be  proved. 

A  theorem  may  be  separated  into  two  parts  : 

(1)  The  hypothesis,  or  what  is  assumed  as  true. 

(2)  The  conclusion,  or  what  is  to  be  proved. 

The  proof  of  a  theorem  is  the  chain  of  reasoning  by  which 
the  conclusion  is  drawn  from  the  hypothesis. 

2.  Theorem.  —  Two  triangles  are  equal  if  a  side  and 
the  two  adjacent  angles  of  one  are  equal,  respectively, 
to  a  side  and  the  two  adjacent  angles  of  the  other. 

HYPOTHESIS. 
Two  A  ABC,  DEF. 
AB  =  DE. 


A  B  D  E 

FIG.  68. 

CONCLUSION.     A  ABC  =  A  DEF. 
PROOF.     Use  the  method  of  superposition. 

(1)  Place  A  ABC  on  DEF  so  that  the  equal  parts  coin- 
cide.    Mention  these  parts,  pair  by  pair. 

(2)  AC  will  coincide  in  direction  with  DF.    Why? 

(3)  BC  will  coincide  in  direction  with  EF.    Why? 

(4)  Therefore  C,  which  is  common  to  AC  and  BC,  must 
fall  on  F,  which  is  common  to  DF  and  EF. 

(5)  Therefore  the  triangles  will  coincide. 

(6)  Therefore  A  ABC  =  A  DEF. 

The  theorem  is  now  proved,  because  the  same  reasoning 
will  hold  good  for  any  two  triangles  that  can  be  drawn. 

3.    Name  all  the  equal  parts  in  the  A  ABC  and  DEF. 


68  LESSONS   IN   GEOMETRY. 

4.  Theorem.  —  In  an  isosceles  triangle  the  angles 
opposite  the  equal  sides  are  equal. 

HYPOTHESIS. 

ABC  an  isosceles  triangle. 

AC  and  BC  the  equal  sides. 

CONCLUSION. 

Z  ABC  =Z  BA C. 

A  D  B 

PROOF.     Successive  steps  :  FIG.  69. 

(1)  Draw  CD  so  as  to  bisect  Z  ACB. 

(2)  Prove   that  A  DBC  =  A  DAC.      They   have   three 
parts  equal,  each  to  each.     CD  is  common.     BC  = AC  by 
hypothesis  ;  Z  BCD  =  Z  ACD  by  construction.    Why,  then, 
are  the  two  triangles  equal  ? 

(3)  The  two  triangles  being  equal,  their  six  parts  are 
equal,  pair  by  pair.     Therefore  Z  ABC  =  Z  BAG. 

5.  What  method  of  proof  is  made  use  of  in  proving 
the  above  theorem? 

The  method  of  equal  triangles.  This  method  may  often 
be  employed  for  the  purpose  of  proving  two  lines  equal,  or 
two  angles  equal.  We  begin  by  finding  or  making  two  tri- 
angles in  which  the  two  lines,  or  the  two  angles,  appear  as 
corresponding  parts.  We  then  prove  these  triangles  to  be 
equal.  Then  it  follows  immediately  that  the  two  lines,  or 
the  two  angles,  must  also  be  equal. 

6.  Apply  this  theorem  to  an  equilateral  triangle. 

In  an  equilateral  triangle  the  angles,  taken  in  pairs,  must 
be  equal,  because  in  each  case  they  are  opposite  equal  sides. 
Therefore  all  three  angles  are  equal. 

An  equilateral  triangle  is  also  equiangular. 


TRIANGLES.  69 

7.  Theorem. — If  two  angles  of  a  triangle  are  equal, 
the  opposite  sides  are  also  equal. 

HYPOTHESIS. 
ABC  a  triangle. 
^ABC=Z.B<AC. 


CONCLUSION.     AC=BC. 

PROOF.     Successive  steps  :  FIG.  70. 

(1)  Draw  CD  so  as  to  bisect  Z  ACB. 

(2)  Prove  that  Z  ADC=  Z  BDC  (see  p.  63,  No.  3). 

(3)  Prove  that  A  ADC=  A  BDC.     Give  reasons  in  full. 

(4)  Therefore  AC  =  BC,  and  the  triangle  is  isosceles. 
What  method  of  proof  is  here  employed  ? 

8.  Compare  the  theorems  in  JVos.  4>  and  7. 
The  hypothesis  of  one  is  the  conclusion  of  the  other. 
When  two  theorems  are  thus  related,  either  one  is  called 

the  converse  of  the  other. 

9.  How  is  the  converse  of  a  theorem  formed  ? 

By  changing  the  hypothesis  to  the  conclusion,  and  the 
conclusion  to  the  hypothesis. 

10.  Apply  the  theorem  of  No.  7  to  a  triangle,  all 
of  whose  angles  are  equal;   that  is  to  say,  an  equi- 
angular triangle. 

If  all  the  angles  are  equal,  then  the  sides  taken  in  pairs 
must  be  equal,  because  each  pair  stands  opposite  equal 
angles.  Therefore  all  the  sides  are  equal. 

An  equiangular  triangle  is  also  equilateral. 

11.  What   is   the  value   in   degrees  of  each  angle  of  an 
equilateral  triangle  ? 


70 


LESSONS    IN    GEOMETRY. 


Lesson  34. 

1.  Theorem.  —  Two  triangles  are  equal  if  the  sides 
of  one  are  equal,  respectively,  to  those  of  the  other. 

HYPOTHESIS. 

In  the  A  ABC,  DEF, 

AB  =  DE, 

AC=DF, 

BC=EF. 

CONCLUSION. 

A  ABC '=  A  DEF. 


FIG.  71. 


PROOF.     Successive  steps  : 

(1)  Place  A  DEF  so  that  the  longest  side  DE  coincides 
with  AB,  and  F  and  C  lie  on  opposite  sides  of  AB. 

(2)  Join  OF.    What  two  isosceles  A  are  thus  formed? 

(3)  Prove  that  Z  ACF=  Z  AFC  (p.  68,  No.  4). 

(4)  Prove  that  Z.BCF=^BFC. 

(5)  Prove  that  Z  ACB=  /.AFB. 

(6)  Prove  that  A  ACB=  A  AFB  (p.  66,  No.  14). 

(7)  Therefore   AACB^ADEF. 

2.  Theorem.  —  Two  right  triangles  are  equal  if  the 
hypotenuse  and  a  leg  of  one  are  equal,  respectively, 
to  the  hypotenuse  and  a  leg  of  the  other. 

HYPOTHESIS. 

Two  right  A  ABC,  DEF; 

AC=EF, 

BC=DF. 


CONCLUSION. 

ABC  =  A  DEF, 


c 


FIG.  72. 


TRIANGLES.  71 

PROOF.     Successive  steps : 

(1)  Place  A  DEF  so  that  the  leg  DF  coincides  with  BC, 
and  A  and  F  are  on  opposite  sides  of  BC. 

What  isosceles  A  is  thus  formed  ?    Why  isosceles  ? 

(2)  Show  that  Z  ACB  =  Z  FCB  (p.  63,  No.  3) . 

(3)  And  that   &ABC=AFBC; 

(4)  That  is,     A  ABC  =  A  DEF. 

3.  Theorem.—^?  perpendicular  is  the  shortest  dis- 
tance from  a  point  to  a  straight  line. 

HYPOTHESIS. 

A  straight  line  AB. 

A  point  P,  not  in  AB. 

PC-LtoAB.  A 

PD  any  other  line  drawn  from 
P  to  AB. 

CONCLUSION.     PC  <  PD. 

"C 

PROOF.     Successive  steps  :  FIG.  73. 

(1)  Produce  PC,  making  CQ  =  PC,  and  join  DQ. 

(2)  Prove  that  A  PDC  =  A  QDC.     Give  all  the  reasons. 
What  two  parts  are  equal  by  construction  ? 

(3)  Then  show  that  PQ  <  PD  +  DQ  (p.  23,  Ax.  9). 

(4)  Hence  show  that  2  PC  <  2  PD. 

(5)  And  therefore  PC  <  PD. 

4,  Define  the  distance  from  a  point  to  a  straight 
line. 

By  the  distance  from  a  point  to  a  line  is  meant  the  shortest 
distance  ;  therefore  the  perpendicular  dropped  from  the  point 
to  the  line. 


72 


LESSONS   IN   GEOMETRY. 


Lesson  35. 

1.  Theorem.  —  Every  point  in  the  perpendicular 
which  bisects  a  straight  line  is  equidistant-  from  the 
ends  of  the  line. 

HYPOTHESIS. 

D  the  middle  point  of  AB. 

CD  _L  to  AB. 

P  any  point  in  CD. 


CONCLUSION.     PA  =  PB.  B 

FIG.  74. 

PROOF.     Prove  that  A  ADP  =  A  BDP.    What  follows  ? 

2,  Theorem.  —  Every  point   equidistant   from  the 
ends  of  a  straight   line   lies  in   the  perpendicular 
which  bisects  the  line. 

HYPOTHESIS. 

D  the  middle  point  of  AB. 

P  any  point  such  that  PA  =  PB. 

CONCLUSION.     PD  is  _L  to  AB. 
PROOF.     Prove   that  A  ADP  = 
A  BDP.  FlG- 75- 

What  follows  with  respect  to  the  angles  ADP  and  BDP? 
Then  show  that  Z  ADP=  90°  (see  p.  48,  No.  2). 

3,  What  truth  follows  from  the  last  theorem? 
Since  two  points  determine  a  straight  line,  any  two  points 

equidistant  from  the  ends  of  a  straight  line  determine  the  per- 
pendicular which  bisects  the  line. 

This  perpendicular  is  called  the  bisecting  perpendicular. 

4.  How  are  the  two  theorems  on  this  page  related  ? 

5.  By  what  method  are  both  theorems  proved? 


TRIANGLES. 


73 


6.  Theorem.  —Every  point  in  the  bisector  of  an 
angle  is  equidistant  from  the  sides  of  the  angle. 

HYPOTHESIS. 

AD  the  bisector  of  Z  BAC. 

P  any  point  in  AD. 

CONCLUSION. 

P  is  equally  distant  from  AB 
and  from  AC.  FIG.  76. 

PROOF.     Successive  steps : 

(1)  Draw  PE  _L  to  AB,  PF  _L  to  AC. 

(2)  Prove  that  A  APE  =  A  APF  (see  p.  63,  No.  3) . 

(3)  What  follows? 

7.  Theorem.— Every  point  within  an  angle,   and 
equidistant  from  its  sides,  lies  in  the  bisector  of  the 
angle. 

HYPOTHESIS. 

P  any  point  within  Z.  BAC  equi- 
distant from  AB  and  AC. 

CONCLUSION. 

PA  bisects  /.  BAC.  FIG.  77. 

PROOF.     Successive  steps : 

(1)  Draw  PE  JL  to  AB,  PF  A.  to  AC.     Join  PA. 

(2)  Prove  that  A  APE  =  A  APF. 

(3)  What  follows? 

8.  What  truth  follows  from  the  last  theorem? 
One  point  equidistant  from  the  sides  of  an  angle  determines 

the  bisector  of  the  angle. 

Why  is  one  point  in  this  case  sufficient  ? 

9.  What  relation  is  there  between  the  theorems  given  on 
this  page?    Why? 


74  LESSONS   IN   GEOMETRY. 

Lesson  36.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  28-31. 

2.  Upon  a  given  length  as  base,  how  many  isosceles  tri- 
angles can  be  constructed  ?     How  many  equilateral  triangles  ? 

3.  If  the  acute  angles  of  a  right  triangle  are  30°  and  60°, 
what  angle  is  formed  by  their  bisectors  ? 

4.  If  one  angle  of  a  triangle  is  40°,  and  the  other  two 
angles  are  equal,  what  are  their  values? 

5.  Find  the  angles  of  a  triangle,  if  the  second  angle  is 
twice  the  first,  and  the  third  three  times  the  second. 

6.  Draw  a  A  ABC,  produce  AB  to  any  point  Z>,  and 
show  that  Z.  DBC  =  Z  BAG  +  Z  ACS. 

7.  Show  that  a  A  with  1,  2,  3  for  sides  cannot  be  made. 

8.  Show  that  a  A  with  1,  2,  4  for  sides  cannot  be  made. 

9.  Two  towns,  A  and  _B,  are  10  miles  apart.     A  place  is 
described  as  7  miles  from  A,  and  5  miles  from  B.     Draw  a 
plan  showing  the  situation  of  (7.     Is  its  position  determined? 

10.  Construct  a  right  triangle  having  one  acute  angle  20° 
larger  than  the  other,  and  hypotenuse  =  4J  in. 

11.  Construct  a  A  ABO ;  given  :  AB  =  2  in.,  Z.  A  =  30°, 
ZC=45°. 

12.  Construct  a  A  ABC',  given:   AB=5  in.,  BC=3% 
in.,  /.  BAC  =  30°,  and  show  that  there  are  two  solutions. 

13.  How  high  is  a  tower  if  the  angle  of  elevation  of  the 
top,  300  yds.  from  the  base,  is  20°? 

14.  A,  J5,  (7,  are  the  corners  of  a  triangular  field.     A  is 
80  ft.  west  of  B,  and  800  ft.  southwest  of  C.     Find  the  cost 
of  fencing  the  field  at  50  cents  a  foot. 


TRIANGLES.  75 

Lesson  37.    Keview. 

1.  Review  the  italicized  exercises  in  Lessons  32-35. 

2.  If  one  side  of  an  equilateral  triangle  is  produced,  what 
is  the  value  of  the  exterior  angle  thereby  formed  ? 

3.  Find  the  values  of  the  angles  at  the  base  of  an  isos- 
celes triangle  if  the  angle  at  the  vertex  is  24°. 

4.  Find  the  angles  of  an  isosceles  triangle  if  each  angle 
at  the  base  is  double  the  angle  at  the  vertex. 

9 

5.  What  are  the  angles  of  an  isosceles  right  triangle? 

6.  Can  you  devise  an  easy  way  to  find  the  height  of  a 
tree  by  means  of  an  isosceles  right  triangle  ? 

7.  Construct  an  isosceles  triangle,  having  given  the  base 
and  the  angle  at  the  vertex.     First  find,  by  construction,  the 
base  angles 

8.  Construct  the  A  ABC,  given  AB=  2  in.,  AC—  3  in., 
BC—  4  in.     Then  find  a  point  D  which  is  2  in.  from  C  and 
equidistant  from  A  and  B. 

9.  Prove  that  the  lines  joining  the  middle  points  of  the 
sides  of  an  equilateral  triangle  divide  the  triangle  into  four 
equal  equilateral  triangles. 

10.  Prove  that  the  altitude  of  an  isosceles  triangle  bisects 
the  base  and  the  angle  at  the  vertex. 

11.  A  monument  AB  on  a  horizontal  plane  subtends  the 
angle  30°  when  seen  from  C  (that  is,  Z  ACB=  30°)  ;  150  ft. 
nearer,  the  angle  it  subtends  is  doubled.     Find  the  height  of 
the  monument. 

12.  A  man  75  ft.  from  the  bank  of  a  stream  finds  that  a 
tree  on  the  opposite  bank  subtends  an  angle  of  45°.     At  the 
bank,  this  angle  is  60°.     How  high  is  the  tree? 


76 


LESSONS  IN   GEOMETEY. 


Lesson  38.    Keview. 

1.  Prove  that  the  construction  in  Fig.  26,  on  page  19,  is 
correct ;  that  is,  prove  that  it  follows  from  the  construction 
that  CF  is  II  to  AB.  For  this  purpose,  show  that  A  DCF= 
ADCE,  and  then  apply  No.  5,  p.  51. 

Prone  the  following  constructions  correct.  In  each  case 
first  state  the  problem ;  then  describe  how  the  construction 
was  made  ;  then  state  what  must  be  proved ;  then  prove  it. 

2.*  Fig.  33,  page  25.  5.    Fig.  48,  page  46. 

3.  Fig.  40,  page  40.  6.    Fig.  49,  page  46. 

4.  Fig.  43,  page  42.  7.    Fig.  50,  page  47. 

8.  Draw  a  A  ABC  and  produce  each  side  in  both  direc- 
tions.    Name  with  letters  all  the  angles  that  are  formed,  and 
find  their  values,  if  Z  ABC  =  70°  and  Z  BAG '=  35°. 

9.  In  general,  three  parts  determine  a  triangle.     Can  you 
think  of  an  exception  to  this  rule  ? 

10.  Find  a  point  in  a  given 
straight   line   AB  which   is 
equidistant  from  two  given 
points  C  and  D.     Prove  that 
your  construction  is  correct. 

11.  Ju    Fig.    78,    AB    is 
given,  and  an  equilateral  tri- 
angle is  constructed,  having 
AB  for  altitude.     The  con- 
struction   is    quite    obvious 
from    the    figure.       CD    is 

drawn  _L  to  AB ;  all  the  arcs  have  the  same  radius. 
Make  the  construction,  and  prove  it  to  be  correct. 


TKIANGLES.  77 


Lesson  39.    Review. 

1 .  Through  each  vertex  of  a  triangle  a  line  parallel  to  the 
opposite  side  is  drawn.     Prove  that  the  triangle  formed  by 
these  lines  is  equal  to  four  times  the  given  triangle. 

2.  Find  the  values  of  all  the  angles  of  an  isosceles  tri- 
angle, if  the  angle  formed  by  producing  the  base  is  130°. 

3.  In  a  A  ABC  the  side  AB  is  produced  to  any  point  D. 
Prove  that  the  Z  CBD  =  Z  BAG  +  Z  ACB. 

4.  Construct  an  isosceles  right  triangle  whose  hypotenuse 
shall  be  6cm  long. 

5.  Describe  the  position  of  all  points  which  are  situated 
2cm  from  a  given  point  A ;    3cm  from  a  given  point  B ;    2cm 
from  A,  and  3cm  from  B. 

6.  A  surveyor  erects  two  flagstaffs,  A,  B,  on  opposite 
sides  of  a  pond,  chooses  a  favorable  point  (7,  and  makes  the 
following  measurements:    .4O=660m,  BC=  1380™,  Z  C '  = 
45°.     Find  the  distance  AB. 

7.  In  a  triangular  field  ABC  a  man  measures  AB=  600m, 
Z  B  =  70°.     He  then  walks  over  BC  without  measuring  it. 
He  finds,   however,  that   Z  (7=45°.      Draw  a   plan,    and 
find  BC. 

8.  Two  roads  meet  at  J.,  forming  the  angle  60°.     How 
far  apart  are  two  houses,  one  of  which  is  situated  700m  from 
A  along  one  road,  and  the  other  900m  from  A  along  the 
other  road? 

9.  The  distances  between  three  towns  A,  J5,  C  are :    AB 
=  9km,  AC=  7km,  BC=  Gkm.     A  straight  railroad  runs  from 
A  to  B.     It  is  desired  to  have  a  station  placed  as  near  as* 
possible  to  C.     Draw  a  plan,  and  find  at  what  distance  from 
A  the  station  should  be  built. 


78 


LESSONS  IN   GEOMETRY. 


Lesson  4O.    Review. 

1.  Re\iew  all  the  italicized  exercises  in  this  chapter. 

2.  Find  in-  one  side  of  a  triangle  a  point  equidistant  from 
the  other  sides.     Prove  your  construction  correct. 

3.  Find  a  point  within  a  triangle  equidistant  from  the 
three  vertices.     Prove  your  construction  correct. 

4.  Find  a  point  within  a  triangle  equidistant   from  the 
three  sides.     Prove  your  construction  correct. 

5.  Three  towns,  J.,  jB,  (7,  are  situated  at  the  vertices  of  an 
isosceles  triangle,  whose  base  is  the  line  BC.    /.  ABC=  30°. 
The  distance  from  A  to  the  base  BC  is  61"11.     Find  the  dis- 
tances between  the  towns. 


X 


FIG.  79. 

6.  In  order  to  find  the  distance  XY  (Fig.  79) ,  without  any 
means  for  measuring  angles,  I  chose  four  points,  A,  B,  (7,  J5, 
and  measured  their  distances  with  the  following  results  : 

AB=140m,  AD=18Qm,  u4(7=115m,  .RD=65m,  CD=85m. 
Draw  a  plan,  and  find  XY. 


CHAPTER  V. 


POLYGONS. 


Lesson  41. 

1.  Define  a  polygon:  its  sides;  perimeter;  vertices. 
A  polygon  is  a  plane  figure  bounded   by  straight  lines. 

The  bounding  lines  are  called  its  sides ;  their  sum,  its  perime- 
ter ;  their  intersections,  its  vertices. 

2.  Name  the  most  important  polygons. 

Polygons  are  named  according  to  the  number  of  sides. 
A  triangle  has  three  sides ;  a  quadrilateral,  four  sides ;  a 
pentagon,  five  sides  ;  a  hexagon,  six  sides  ;  an  octagon,  eight 
sides  ;  a  decagon,  ten  sides  ;  a  dodecagon,  twelve  sides. 

3.  Define  a  diagonal,  and  explain  how  a  polygon 
may  be  divided  into  triangles  by  drawing  diagonals. 

A  diagonal  is  a  straight  line  joining  two  vertices  not  on 
the  same  side.  A  polygon  may  be  divided  into  triangles  by 
drawing  all  the  diagonals  possi- 
ble from  one  vertex.  The  num- 
ber of  diagonals  that  can  be 
drawn  from  a  vertex  is  always 
three  less  than  the  number  of 
sides ;  and  the  number  of  tri- 
angles into  which  the  polygon  is 
divided  is  always  two  less  than 
the  number  of  sides. 


80  LESSONS   IN   GEOMETRY. 

4.  Theorem.  —  The  sum  of  the  angles  of  a  polygon  is 
equal  to  180°  X  the  number  of  sides  less  two. 

We  prove  this  theorem  for  a  hexagon  as  follows : 

HYPOTHESIS. 

A  hexagon  ABCDEF. 

CONCLUSION. 

The  sum  of  the  A  of  the 
hexagon  is  equal  to  180°  x  4, 
or  720°. 

PROOF.  Divide  the  hexagon  into  6  —  2,  or  4,  triangles,  by 
drawing  the  diagonals  AC,  AD,  AE. 

Obviously,  the  sum  of  the  A  of  the  hexagon  is  equal  to 
the  sum  of  the  A  of  these  triangles. 

The  sum  of  the  A  of  the  A  =  180°  x  4.    Why? 

Therefore  the  sum  of  the  A  of  the  hexagon  =  180°  x  4. 

Now  suppose  that  in  place  of  6  sides  the  polygon  has  any 
number  n  of  sides  ;  then  there  would  be  n  —  2  triangles  in 
place  of  4  ;  and,  therefore,  the  sum  of  the  A  of  the  polygon 
would  be  180°x(n-2). 

5.  Prove,  without  assuming  the  above  theorem  to  be  true, 
that  the  sum  of  the  angles  of  a  quadrilateral  is  equal  to  360°. 

6.  If  the  angles  of  a  quadrilateral  are  all  equal,  what  is 
the  value  of  each  one? 

7.  Three  angles  of  a  quadrilateral  are  40°,  70°,  and  120°. 
What  is  the  value  of  the  other  angle  ? 

8.  Find  the  sum  of  the  angles  of  a  pentagon. 

9.  Find  the  sum  of  the  angles  of  an  octagon. 

10.  Find  the  sum  of  the  angles  of  a  decagon. 

11.  Find  the  suin  of  the  angles  of  a  dodecagon. 


POLYGONS. 


81 


11.  Classify  quadrilaterals. 

Quadrilaterals  are  divided   into  three  classes :   parallelo- 
grams, trapezoids,  and  trapeziums. 

Parallelograms  have  their  opposite  sides  parallel. 
Trapezoids  have  only  two  sides  parallel. 
Trapeziums  have  no  parallel  sides. 


PARALLELOGRAM. 


TRAPEZOID. 


TRAPEZIUM. 


12.  Classify  parallelograms. 

There  are  four  kinds  of  parallelograms :  the  square,  the 
rectangle,  the  rhombus,  and  the  rhomboid. 

A  square  has  four  equal  sides  and  four  right  angles. 

A  rectangle  has  unequal  adjacent  sides  and  four  right 
angles. 

A  rhombus  has  four  equal  sides  and  no  right  angles. 

A  rhomboid  has  unequal  adjacent  sides,  and  no  right 
angles. 


SQUARE. 


RECTANGLE. 


RHOMBUS. 


RHOMBOID. 


13.  Make,   freehand,   a   parallelogram ;    a   trapezoid ;    a 
trapezium. 

14.  Make,  freehand,  a  trapezium  having  two  right  angles. 

15.  Give  examples  of  parallelograms.    What  kind  is  each  ? 

16.  Make,  freehand,  a  square  ;    a  rhombus  ;    a  rectangle  ; 
and  a  rhomboid. 


82  LESSONS  IN   GEOMETRY. 


Lesson  42. 

NOTE.  —  In  working  the  exercises  in  this  lesson,  construct  perpen- 
diculars and  parallels  with  the  ruler  and  the  triangle,  and  angles  with 
the  protractor. 

1.  How  are  the  dimensions  of  a  parallelogram 
named  ? 

They  are  called  base  and  altitude.  Any  side  may  be  taken 
as  the  base,  and  then  the  altitude  will  be  the  perpendicular 
dropped  to  the  base  from  any  point  of  the  opposite  side. 
Thus,  in  Fig.  82,  AB  is  the  base,  DE  the  altitude. 


A          E  B  A  E  B 

FIG.  82.  FIG.  83. 

2.  How  are  the  dimensions  of  a  trapezoid  named  ? 

They  are  also  named  base  and  altitude.  One  of  the  par- 
allel sides  is  always  taken  as  the  base,  and  the  altitude  is 
the  perpendicular  drawn  from  one  parallel  side  to  the  other. 
Thus,  in  the  trapezoid  ABCD  (Fig.  83)  DE  is  the  altitude. 

3.  How  are  the  sides  of  a  trapezoid  named  ? 

The  parallel  sides  are  called  the  bases,  and  the  other  two 
sides  are  called  the  legs. 

4.  Construct  a  parallelogram  with  a  base  of  3  in.  and  an 
altitude  of  2  in.  ;  also  a  trapezoid  with  the  same  dimensions. 

5.  Draw  a  triangle  ABC.     Through  D,  any  point  in  AC, 
draw  DE  II  to  AB,  and  DF  II  to  BC.    What  kind  of  a  figure 
is  ABED  ?   Why  ?   What  kind  of  a  figure  is  BEDF  ?   Why  ? 


POLYGONS. 


83 


6.  Construct  a  square  having  a  side  2  in.  long.     Draw 
its  diagonals.    Measure  their  lengths  and  the  angle  between 
them. 

7.  Construct  a  rhombus  with  each  side  2  in.  long,  and 
one  angle  equal  to  60°.     Draw  its  diagonals.     Measure  their 
lengths  and  the  angle  between  them. 

8.  Construct  a  rectangle  having  for  two  adjacent  sides 
l^in.  and  2  in.     Draw  its  diagonals.     Measure  their  lengths 
and  the  angle  between  them. 

9.  Construct  a  rhomboid  ABCD,  having  given  AB=  3^ 
in.,  £<7=l£in.,  Z.  ABC =55°.      Find   the  values  of   the 
other  three  angles. 

10.  Draw  a  line  AB ;  upon  one  side  construct  an  equi- 
lateral A  ABC,  upon  the  other  side  a  square  ABDE.     Join 
CD,  CE.     Prove  that  A  CDE  is  isosceles. 

11.  Construct  two  rectangles,  one  within  the  other,  and 
make  their  adjacent  sides  everywhere  -J-  of  an  inch  apart. 

12.  Construct  two  squares.     Divide  one  into  four  equal 
squares,  and  the  other  into  four  equal  rectangles. 

13.  Construct   Fig.    84    (scale,    2:1). 
Then  erase  the  dotted  lines,   and   shade 
the  portions  of  surface  at  the  corners  and 
at  the  middle  of  the  sides. 

14.  Through  a  point  0  draw  two  per- 
pendicular lines.    With  centre  0  and  any 
radius,  describe  a  circle,  cutting  the  lines 
in  A,  B,  C,  D.     Join  AB,  BC,  CD,  DA. 

What  kind  of  a  figure  is  ABCD?    Can  you  prove  your 
answer  to  be  correct? 

15.  A  man  walks  5  miles  due  north,  then  3  miles  north- 
west, then  4  miles  due  south,   and  then  he   goes  straight 
home.    What  figure  does  his  path  enclose? 

Find,  by  drawing  to  scale,  how  far  he  has  walked. 


FIG.  84. 


LESSONS   IN   GEOMETRY. 


Lesson  43. 

1.   Theorem.  —  The  opposite  angles  of  a  parallelo- 
gram are  equal. 

HYPOTHESIS. 

A  parallelogram  ABCD. 

Also  let  its  angles  be  de- 
noted by  a,  6,.c,  d,  as  shown 
in  the  figure. 

CONCLUSION,     a  =  c,  b  =  d. 


E 


FlG.  85. 


PROOF.     Produce  AS  to  2£,  DC  to  F,  and  let 


We  know  that  a  =  m  (p.  50,  No.  3). 

And  that  c  =  m  (p.  51,  No.  4). 

Therefore  a  =  c  (p.  23,  Ax.  1). 

Also,  we  know  that  d  =  n.  Why? 

And  that  b  =  n.  Why? 

Therefore  b  =  d.  Why? 

2.  Theorem.  —  The  opposite  sides  of  a  parallelogram 
are  equal. 

What  is  the  hypothesis?  D 

What  is  the  conclusion  ? 

PROOF.    Successive  steps : 

(1)  Draw  the  diagonal  AC. 

(2)  Prove  that   A  ABC  — 
A  ADC.    What  follows? 


FIG.  86. 


3.   What  truths  follow  immediately  from  No.  2? 

(1)  A  diagonal  divides  a  parallelogram  into  two  equal  A. 

(2)  Parallels  between  parallels  are  equal. 


POLYGONS.  85 

4.  Theorem.  —  Two  parallels  are  everywhere  equi- 
distant. 

HYPOTHESIS.  £ — ? § — 

AB  is  II  to  CD. 

E,  Gr,  any  points  in  AB. 

EF  is  _L  to  AB. 

GH  is  _L  to  AB.  ^ — ^—               — 5- 

CONCLUSION.    EF=GH.  FlGt  87> 


PROOF.     ^FEG  =  /.HGB.    Why? 
Then  JS^  is  II  to  <7#  (p.  51,  No.  5). 
Hence  EF=GH.     Why? 

5.  One  angle  of  a  parallelogram  =  50°.    Find  the  others. 

6.  What  is  the  angle  which  the  diagonal  of  a  square 
forms  with  one  of  the  sides? 

7.  In  what  kinds  of  parallelograms  is  the  altitude  equal 
to  one  of  the  sides  ? 

8.  How  many  trees  10  yds.  apart  can  be  set  out  around 
a  rectangular  park  800  yds.  long  and  480  yds.  wide? 

9.  Upon  a  line  AB  as  side,    construct  with  ruler  and 
compasses  a  square.     Draw  BC  _L  to  AB  ;  take  BC  —  AB  ; 
and  with  centres  A  and  0,  and  radius  AB,  describe  arcs 
meeting  at  D.     ABCD  is  the  required  square.     Prove  that 
it  is  a  square. 

10.  Construct  a  square  having  its  perimeter  equal  to  a 
line  MN. 

11.  Construct  a  square  having  its  diagonal  equal  to  a 
line  MN. 

12.  Make  a  plan  of  a  wall  20  ft.  high  and  64  ft.  long, 
with  a  passage-way  in  the  middle  10  ft.  high  and  16  ft.  wide. 

13.  Make  a  plan  of  two  vertical  posts  18  ft.  high  and 
24  ft.  apart,  with  a  cross-bar  10  ft.  from  the  ground. 


86 


LESSONS   IN   GEOMETRY. 


Lesson   44. 

1,    Theorem.  —  The  diagonals  of  a  parallelogram 
bisect  each  other. 


HYPOTHESIS. 

Let  ABCD  be  a  parallelo- 
gram, and  let  the  diagonals 
AC,  BD,  intersect  at  0. 

CONCLUSION. 


FIG.  88. 


PROOF.     Prove  that  A  ABO  =  A  DCO,  and  keep  in  mind 
that  in  equal  triangles  equal  sides  are  opposite  equal  angles. 

2.  Prove   that   the  diagonals  of  a  rectangle   are   equal. 
Begin  by  stating  the  hypothesis  (Fig.  89)  ;    then  state  the 
conclusion  ;  then  prove  by  showing  that  the  triangles  ABD, 
ACD,  are  equal. 

3.  Prove  that  the  diagonals  of  a  rhombus  are  perpen- 
dicular to  each  other  (Fig.  90) .     State  hypothesis  and  con- 
clusion, and  prove  by  showing  that  A  AOB  =  AAOD,  etc. 

C 


FIG.  89. 

4.  Prove  that  the  construction,  on  page  25,  for  dividing  a 
line  into  equal  parts,  is  a  correct  one. 

Draw  CM,  DN,  etc.,  II  to  AB,  prove  the  A  AHC,  CMD, 
etc.,  equal,  and  apply  No.  3,  p.  84,  and  Ax.  1,  p.  23. 


POLYGONS.  87 

5.  Problem.  —  To  construct  a  parallelogram,  having 
given  two  adjacent  sides  and  the  included  angle. 

CONSTRUCTION.     Instruments  :  ruler  and  compasses. 

Let  a  and  b  be  the  given  sides  ;  ra  the  given  angle. 

Make  Z  BAD  =  m. 

Lay  off  AB  =  a,  AD  =  b.  ' j£~~ 

With  centre  D  and  radius  a 
describe  an  arc  EF. 

With  centre  B  and  radius  b 
describe  an  arc  cutting  EF  at  (7. 

Join  BC  and  DC.  ^  9i 

ABCD  is  the  figure  required. 

PROOF.    (1)  The  figure  obviously  has  the  three  given  parts. 

(2)  Join  .4(7.     A  ABC  =  A  ADC.    Why? 

(3)  Therefore  Z  BAC  =  Z  DC  A,  and  Z  .4<7£=  Z  C^Z>. 

(4)  Therefore  AB  is  II  to  DC,  and  £C  is  II  to  AD.    Why? 

(5)  Therefore  ABCD  is  a  parallelogram.    Why? 

NOTE.  —  In  the  following  exercises,  use  ruler  and  compasses  only. 
First  make  the  construction ;  then  describe  it ;  then  prove  it  to  be  correct. 

6.  Construct  a  square,  having  given  a  side. 

7.  Construct  a  square,  having  given  the  diagonal. 

8.  Construct  a  rectangle,  given  two  adjacent  sides. 

9.  Construct  a  rectangle,  given  a  side  and  a  diagonal. 

10.  Construct  a  rhombus,  given  the  two  diagonals. 

11.  Construct  a  rhombus,  given  a  side  and  an  angle. 

12.  Construct  a  parallelogram,  given   two   sides   and   a 
diagonal. 

13.  Construct  a  parallelogram,  given  one  side,  one  angle, 
and  one  diagonal. 

14.  Construct  a  trapezoid,  given  the  bases,  the  altitude, 
and  one  of  the  legs. 


LESSONS   IN   GEOMETRY. 


Lesson  45. 

1.   Problem.  —  To  construct  a  polygon  equal  to  a 
given  polygon  ABCDEF  (Fig.  92). 

CONSTRUCTION.     Instruments  :  ruler  and  compasses. 
Draw  the  diagonals  AC,  AD,  AE. 

Make  A  MNO  =  A  ABC  (explain  fully  how) . 

Then  make  A  MOP  =  A  ACD. 

Also  A  MPQ  =  A  ADE. 

And  A  MQR  =  A  AEF. 

Then  the  polygon  MNOPQ  R  is  the  polygon  required. 
ED  Q          P 


B 


N 


FIG.  92. 


PROOF.     By  construction,  the  A,  pair  by  pair,  are  equal. 
They  are  also,  pair  by  pair,  similarly  placed. 

Therefore  the  polygon  MNOPQR=the  polygon  ABCDEF. 

NOTE.  —  Use  only  ruler  and  compasses  for  the  following  exercises. 
First  make  the  construction ;  then  describe  it ;  then  prove  it  to  be  correct. 

2.  Construct  a  square,  given  the  perimeter. 

3.  Construct  a  rectangle,  given  one  side  and  the  angle 
between  the  diagonals. 

4.  Construct  a  rectangle,  given  a  diagonal  and  the  angle 
between  the  diagonals. 

5.  Construct  a  rectangle,  given  one  side  and  the  perimeter. 

6.  Construct  a  rhombus,  given  one  side  and  the  altitude. 


POLYGONS.  89 

7.  Construct  a  rhombus,  given  a  side  and  a  diagonal. 

8.  Construct  a  parallelogram,  given  the  base,  the  alti- 
tude, and  one  angle. 

9.  Construct  a  parallelogram,  given  the  two  diagonals 
and  the  angle  between  them. 

10.  Construct  a  parallelogram,  given  the  sides,  and  know- 
ing that  one  angle  is  double  the  other. 

11.  Construct  a  trapezoid,  given  the  legs,  one  base,  and 
the  altitude. 

12.  Construct  a  trapezoid,  given  one  leg,  one  of  the  bases, 
and  the  angles  at  the  base. 

13.  Construct  a  trapezium,  given  the  four  sides  and  one 
of  the  angles. 

14.  Construct  a  trapezium,  given  the  four  sides  and  one  of 
the  diagonals. 

15.  Make  a  pentagon,  and  then  construct  an  equal  pen- 
tagon. 

16.  Make  a  hexagon,  and  then  construct  an  equal  hexagon. 

17.  Make  an  octagon,  and  then  construct  an  equal  octagon. 

18.  Draw  any  triangle  ABC.      Upon  AB  describe   the 
square  ABEF,  and  upon  AC  the  square  ACGH.     Drop  a 
perpendicular  from  A  to  BC,   and  join  EC,  GB.     If  the 
figure  is  correctly  drawn,  EC  and  GB  will  intersect  on  the 
perpendicular. 

19.  Draw  a  cross  vertical  section  of  a  ditch  whose  depth 
'is  6  ft.,  breadth  at  bottom  4  ft.,  at  top  8  ft.,  and  whose  sides 

are  equal.     Find  the  length  of  the  side.     In  this  exercise, 
besides  ruler  and  compasses,  the  scale  must  be  used. 

20.  Construct  a  square,  and  on  the  four  sides  construct 
equilateral  triangles ;  join  their  vertices,  and  show  by  meas- 
urement that  the  figure  so  formed  is  also  a  square.     Can  you 
prove  that  it  is  a  square  ? 


90  LESSONS   IN  GEOMETRY. 

Lesson  46. 

1.  Define  an  equilateral  polygon. 

An  equilateral  polygon  is  a  polygon  whose  sides   are 
equal. 

2.  Define  an  equiangular  polygon. 

An  equiangular  polygon  is  a  polygon  whose  angles  are 
equal. 

3.  Define  a  regular  polygon. 

A  regular  polygon  is  a  polygon  which  is  both  equilateral 
and  equiangular. 

4.  How  is  an  angle  of  a  regular  polygon  found  ? 
First  find  the  sum  of  all  the  angles  (p.  80,  No.  4)  ;  then 

divide  this  sum  by  the  number  of  angles  ;  in  other  words,  by 
the  number  of  sides  of  the  polygon. 

5.  What  are  the  usual  names  of  the  regular  triangle  and 
the  regular  quadrilateral  ? 

6.  What  is  the  perimeter  of  a  regular  octagon,  if  one  side 
is  6  in.  long? 

7.  What  is  the  length  of  one  side  of  a  regular  decagon, 
if  the  perimeter  is  35  in.  ? 

8.  What  quadrilateral  is  equilateral,  but  not  equiangular? 
also  what  quadrilateral  is  equiangular,  but  not  equilateral? 

9.  Make  a  pentagon  which  shall  be  equilateral,  but  not 
equiangular. 

10.  Make  a  hexagon  which  shall  be  equiangular,  but  not 
equilateral. 

11.  Find  the  angle  of  a  regular  dodecagon. 

12.  Find  the  angles  of  the  regular  polygons  from  three 
sides  to  ten  sides,  and  arrange  the  results  in  tabular  form. 


POLYGONS. 


91 


As  the  number  of  sides  increases,  in  what  way  does  the 
angle  change? 

13.  Stone  pavements  furnish  examples  of  the  use  of  regu- 
lar polygons.     Only  those  polygons  can  be  used  which  will 
fill  up  all  the  space  about  a  point. 

Show  that  equilateral  triangles  may  be  used,  and  make  a 
pattern  like  that  in  Fig.  93. 

14.  Make  a  pattern  arranged  in  squares,  like  that  shown 
in  Fig.  94. 


FIG.  93. 


FIG.  94. 


FIG. 95. 


15.  Show  that  a  pattern  can  be  made  of  regular  hexagons 
arranged  in  groups  of  three  about  a  point,  and  make  a  pat- 
tern like  that  in  Fig.  95. 

16.  Make  a  pattern  consisting  of  a  combination  of  equi- 
lateral triangles  and  hexagons  (Fig.  96). 


FIG.  96. 


FIG.  97. 


17.  Show  that  a  pattern  cannot  be  made  of  regular  octa- 
gons alone,  but  that  one  can  be  made  of  regular  octagons 
combined  with  squares,  as  shown  in  Fig.  97. 


92  LESSONS   IN   GEOMETRY. 

Lesson  47.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  41-43. 

» 

2.  A  diagonal  divides  a  parallelogram  into  two'equal  tri- 
angles.   What  kind  of  triangles  are  they  if  the  parallelogram 
is  a  rectangle  ?  a  rhombus  ?  a  square  ? 

3.  The  perimeter  of  a  rectangular  lot  is  360  ft.,  and  the 
street  side  is  80  ft.     Find  the  depth  of  the  lot. 

4.  Make,   freehand,   a  hexagon  having   as  many  right 
angles  as  possible. 

5.  Find  the  sum  of  the  angles  of  a  polygon  of  15  sides. 

6.  Prove  that  the  figure  formed  by  joining  the  middle 
points  of  the  sides  of  a  square,  taken  in  order,  is  also  a 
square. 

7.  Draw  a  triangle  ~ABC.     Through  B  and  C  draw  par- 
allels to  the  opposite  sides,  meeting  at  D.    What  kind  of  a 
figure  is  ABCD?    Why? 

8.  A  man  walks  8  miles  east;  then  4|-  miles  southwest; 
then  8  miles  west ;  then  straight  home.     Draw  a  plan,  and 
find  how  far  he  walked. 

9.  Construct  a  trapezoid  having  an  altitude  of  1  in.,  and 
equal  legs  each  2  in.  long. 

10.  Construct  a  quadrilateral  ABCD,   given  AE—AD 
=  2  in.,  BC=  CD  =  1  in.,  Z  BAD  =  35°. 

11.  A  man  walks  3  miles  ;  turns  to  his  right  through  60°, 
and  goes  2  miles  ;  turns  again  to  his  right,  through  60°,  and 
goes  2  miles  ;  and  then  goes  straight  home.     Draw  a  plan, 
and  find  how  far  he  has  walked. 

12.  Construct  a  square  ABCD,  and  upon  its  sides  equi- 
lateral triangles  ABE,  BCF,  CDG.     Join  E,  F,  G,  H,  taken 
in  order.    What  kind  of  a  triangle  is  AEH?    What  kind  of 
a  figure  is  EFGH1    Prove  your  answers, 


POLYGONS.  93 

Lesson  48.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  44-46. 

2.  Into  what  figures  is  a  rhombus  divided  by  drawing 
both  its  diagonals? 

3.  Construct  a  rhombus,  one  side  of  which  shall  be  2  in. 
long,  and  one  angle  of  which  shall  be  double  the  other. 

4.  The  intersection  of  the  diagonals  of  a  parallelogram  is 
often  called  its  centre.     Prove  that  any  straight  line  which 
can  be  drawn  through  the  centre  will  divide  the  perimeter 
into  two  equal  parts. 

5.  Construct  a  square,  given  one  side  and  the  position  of 
its  centre. 

6.  Construct  a  trapezoid,  given  one  base,  the  altitude, 
and  the  two  diagonals. 

7.  A  stone  dam  is  20  ft.  high,  34  ft.  wide  at  the  bottom, 
and  4  ft.  wide  at  the  top.     The  slant  height  of  each  side  is 
the  same.     Draw  to  scale  a  cross  section  of  the  dam,  and 
find  the  slant  height. 

8.  Construct  a  parallelogram,  given  one  side  and  the  two 
diagonals. 

9.  A  man  walks  5  miles  straight ;  then  turns  through  30°, 
and  walks  2  miles  ;  then  takes  a  direction  parallel,  but  oppo- 
site, to  his  first  direction.     Draw  a  plan  of  his  route.     How 
far  is  he  from  his  starting-point? 

10.  Draw  two  lines  making  an  angle  of  60°,  and  find  a 
point  which  shall  be  1  in.  distant  from  one  line,  and  2  in. 
distant  from  the  other. 

11.  A  certain  town  is  3  miles  from  a  straight  railroad.     A 
man  lives  5  miles  from  the  town,  and  1  mile  from  the  rail- 
road.    Find,  by  construction,  how  many  locations   satisfy 
these  conditions. 


94  LESSONS   IN   GEOMETBY. 

Lesson  49.     Review. 

1.  Construct  a  square,  given  the  diagonal  =  7cm. 

2.  Construct  a  rhombus  ABCD,  given  AB=5cm,  Z  ABC 
=  45°. 

3.  Construct  a  rectangle  ABCD,  given  AB  =  4cm,  and 
the  diagonal  AC=  9cm. 

Construct  a  parallelogram  ABCD,  having  given : 

4.  AB  =  3cm,         AD  =  5cm,  Z  ABC  =70°. 

5.  AB  =  4cm,         AD=7cm,          AC=9cm. 
6     AC  =  7cm    Z  ACD 30°    Z  DAC1  =  50° 

7.  AB=6cm,          BD=8cm,  ZABD=30°. 

8.  ^B=4cm,  Z  CAB  =  80°,  Z  DAB  =100°. 

Construct  a  trapezoid  ABCD  (AB  II  to  CD) ,  having  given  : 

9.  AB=3cm,        CD—5cm,        AD=4:cm,    ZADC=  70°. 

11.  BD=Qcm,ZDBC=QO°,ZBDC=  30°,  ZADB=  30°. 

12.  AB=4:cm,       BD=5cm,        BC=3cm,    Z.CBD=  80°. 

Construct  a  quadrilateral  ABCD,  having  given : 

13.  AB=3cm,  AC=5cm,  BD=±cm, 

14.  AB=3cm,  AC=4:cm,  BD=5cm, 

15.  Draw  a  plan  of  a  rectangular  floor  15m  long  and  8m 
wide,  and  find  the  distance  between  the  opposite  corners. 

16.  ABCD   is   a   rectangle.      JLB=1200m,   BC=  600m. 
What  is  the  distance  from  A  to  the  middle  point  M  of  the 
side  CD?    What  kind  of  a  triangle  is  ABM?    Prove  your 
answer. 

17.  A  barn  is  12m  wide,  8m  high  to  the  eaves,  and  14m  high 
to  the  ridgepole.     Draw  a  plan  of  a  vertical  cross  section. 
How  long  are  the  rafters? 


CHAPTER  VI. 


THE   CIRCLE. 


Lesson  5O. 

1.  Review  the  italicized  exercises  in  Lesson  7. 

2.  Define  an  angle  at  the  centre,  a  sector,  a  seg- 
ment, a  quadrant,  a  semicircle. 

An  angle  at  the  centre  is  an  angle  whose  vertex  is  at  the 
centre  of  a  circle.  (For  example  :  Z.  AOB,  Fig.  98.) 

A  sector  is  a  portion  of  a  circle  bounded  by  two  radii  and 
the  intercepted  arc. 

A  segment  is  a  portion  of  a  circle  bounded  by  an  arc  and 
its  chord. 

A  quadrant  is  a  sector  whose  radii  form  a  right  angle. 

A  semicircle  is  a  sector  whose  radii  form  a  straight  angle, 
or  segment  whose  chord  is  a  diameter  of  the  circle. 

3.  Define  a  semi-circumference. 

4.  Compare,  as  regards  size,  a  circle, 
a  semicircle,  and  a  quadrant. 

5.  Draw  a  circle,  and  make  in  it  an 
angle  at  the  centre ;  a  sector ;  a  seg- 
ment ;   a  quadrant.      Also  name  them 
with  letters. 

6.  Into  what  figures  is  a  sector  divided  by  the  chord  of 
its  arc? 

7.  Construct  a  sector  whose  angle  is  60°. 

8.  Construct  a  segment  whose  chord  =  radius  of  the  circle. 


90  LESSONS   IN   GEOMETRY. 

9.  Define  equal  circles  and  equal  circumferences. 
Two  circles,  or  two  circumferences,  are  equal  if  they  can 

be  so  placed  that  they  coincide. 

10.  What  truth  follows  immediately  from  the  defi- 
nitions which  have  been  given  ? 

If  two  circles  have  equal  radii,  or  equal  diameters,  they 
are  equal ;  and  their  circumferences  are  also  equal. 

11.  Define  equal  arcs  (p.  44>  No.  6). 

12.  What  conditions  must  two  equal  arcs  satisfy? 

(1)  They  must  have  equal  radii. 

(2)  They  must  correspond  to  equal  angles  at  the  centre. 

13.  Draw  two  unequal  arcs  having  equal  radii. 

14.  Draw   two   unequal    arcs    that   correspond  to   equal 
angles  at  the  centre.     (The  radii  must  be  unequal.) 

15.  Theorem.  — JT&  equal  circles,  equal  angles  at  the 
centre  intercept  equal  arcs  (Fig.  99). 

What  is  the  hypothesis  ? 
What  is  the  conclusion  ? 

PROOF.      Place    /.  AOB    on 
Z  CRD  so  that  they  coincide. 

A  will  fall  on  C.     Why?  ^ 

B  will  fall  on  D.     Why? 

Arc  AB  will  coincide  with  arc  CD  because  all  their  points 
are  at  equal  distances  from  R. 

16.  Theorem.  —  In  equal  circles,  equal  arcs  corre- 
spond to  equal  angles  at  the  centre  (Fig.  99). 

What  is  the  hypothesis  ?     What  is  the  conclusion  ? 
Prove  by  the  method  of  superposition  (as  in  No.  15) . 

17.  In  what  relation  does  No.  16  stand  to  No.  15? 


THE  CIRCLE.  97 

18.  In  what  way  is  the  term  "  subtend"  used? 

A  chord  AB  (Fig.  100)  is  said  to  subtend  the  arc  AB,  and 
the  arc  AB  is  said  "to  be  subtended  by"  the  chord  AB. 
To  subtend  means  to  stretch  across. 

19,  Theorem.—  In  equal  circles,  equal  chords  sub- 
tend equal  arcs  (Fig.  100). 

What  is  the  hypothesis  ? 
What  is  the  conclusion  ? 


PROOF.     Apply  No.  1,  p.  70, 
and  No.  15,  p.  96.  FIG.  100. 

20.  Theorem.  —  /^  equal  circles,  equal  arcs  are  sub- 
tended by  equal  chords  (Fig.  100). 

What  is  the  h}*pothesis? 
What  is  the  conclusion  ? 

PROOF.     Apply  No.  16,  p.  96,  and  No.  14,  p.  66. 

21.  What  relation  is  there  between  Nos.  19  and  20? 

22.  Problem.  —  To  make  an  arc  equal  to  a  given  arc 
AB  whose  centre  is  at  a  given  point  0  (Fig.  101). 

CONSTRUCTION.  Draw  OA, 
OB,  AB. 

With  radius  OA,  and  any 
point  R  as  centre,  describe  an 
arc  CE.  /  \  \ 

With  centre   (7,  and  radius        /  \     \    / 

equal  to  AB,  cut  CE  at  D.  ^  ^    V 

Then  arc  CD  =  arc  AB.  FIG.  101. 

PROOF.     Apply  No.  19. 

23.  Make  an  arc  equal  to  three  times  a  given  arc. 


LESSONS   IN   GEOMETRY. 


Lesson  51. 

1.  Theorem.  —  The  radius  perpendicular  to  a  chord 
bisects  the  chord,  the  arc  subtended  by  the  chord,  and 
the  corresponding  angle  at  the  centre. 

HYPOTHESIS. 

Let  the  radius  00  be  J_  to  the 
chord  AB,  and  meet  AB  at  0. 

CONCLUSIONS. 

AD=BD, 
arc  AC  '=  arc  J50, 


PROOF.    Compare  A  AOD,  BOD,  and  apply  No.  16,  p.  96. 

2.  Theorem.  —  The  perpendicular  erected  at  the 
middle  point  of  a  chord  passes  through  the  centre  of 
the  circle. 


HYPOTHESIS. 

Let  AB  be  a  chord. 

Let  D  be  its  middle  point. 

Also  let  DE  be  J_  to  AB. 

CONCLUSION. 

DE  passes  through  0,  the  centre  of 
the  circle. 


FIG.  103. 


PROOF.     DE  passes  through  all  points  which  are  equally 
distant  from  A  and  B.     (No.  1,  p.  72.) 

0  is  a  point  equally  distant  from  A  and  B.    Why  ? 
Therefore  DE  passes  through  0. 

3.  Bisect  an  arc,  and  prove  your  construction  correct. 

4.  Construct  an  arc  containing  22°  30'. 


THE  CIRCLE.  99 

5.  Problem.  —  To  describe  a  circumference  through 
three  points,  A,  B,  C,  not  in  a  straight  line. 

CONSTRUCTION". 

Draw  AB,  EG. 

Draw  the  bisecting  perpendiculars  to 
AB  and  BC. 

Let  these  _k  meet  at  0. 

With  centre  0  and  radius  OA  de- 
scribe a  circle. 

This  circle  will  pass  through  the 
points  A,  B,  and  C. 

PROOF.     Apply  No.  1,  p.  72,  or  No.  2,  p.  98. 

6.  Can  a  circle  be  described  through  three  points  which 
are  in  the  same  straight  line  ? 

7.  Describe  a  circle  and  erase  the  centre.     Then   find 
the  centre  by  a  construction,  and  prove  that  your  construc- 
tion is  correct.     (Apply  No.  2,  p.  98.) 

8.  Draw  a  triangle,   and  then  describe  a  circle  which 
shall  pass  through  its  vertices. 

9.  Make   a  square  of  2  in.  side,  and  describe  a  circle 
which  shall  pass  through  its  vertices. 

10.  Describe  three  circles,  all  passing  through  two  points 
A  and  B.     How  many  circles  can  be  described  through  two 
given  points  ?    In  what  line  are  all  the  centres  located  ? 

11.  Describe  a  circle  with  a  radius  of  2  in.  which  shall 
pass  through  two  points  A  and  jB,  3  in.  apart. 

12.  Mark  two  points  A  and  B,  and  draw  a  line   CD. 
Then  describe  a  circle  whose  centre  shall  be  in  the  line  OZ>, 
and  whose  circumference  shall  pass  through  both  A  and  B. 
Can  you  draw  CD  so  that  the  problem  cannot  be  solved  ? 


100 


LESSONS   IN   GEOMETRY. 


Lesson  52. 

1.  Define  an  inscribed  angle. 

An  inscribed  angle  is  an  angle  whose  vertex  is"  in  the  cir- 
cumference of  a  circle,  and  whose  sides  are  chords. 

For  example  :  Z  SAC  (Fig.  105)  is  an  inscribed  angle. 

2.  Theorem.  —  An  inscribed  angle  is  equal  to  half 
the  angle  at  the  centre,  which  intercepts  the  same  arc. 

HYPOTHESIS.    Circle  with  centre  0.    An  inscribed  Z  BAG. 
CONCLUSION.    BAC  =  |  BOO. 

Case  1  (Fig.  105).     The  centre  0  in  the  line  AB. 
PROOF.   ZBAC  +  ZOCA   +  Z^OC=180°.    Why? 
Also  Z  BAG  =  Z  OCA.    Why  ? 

Therefore  2 Z  BAG  +  Z  AOC  =180°.    Why? 

But  ^BOC  +  Z  AOC  =  ISO0. 

Hence  2 ZJB.4C  +  Z  .40(7  =  ZBOC  +  ZAOC.    Why? 
Therefore  2^  BAG  =  /.BOG.  (Ax.  3.) 

Therefore  Z  BAG  =|Z  BOG.  (Ax.  5.) 

A  A  A 


FIG.  107. 


Case  2  (Fig.  10G).     The  centre  0  between  AB  and  AC. 
Draw  the  diameter  AD.    Apply  Case  1.     Add  the  results. 
Case  3  (Fig.  107).     The  centre  0  outside  the  angle  BAG. 
Draw  AD.     Apply  Case  1.     Subtract  the  results. 


THE   CIRCLE. 


101 


3.  When  is  an  angle  inscribed  in  a  segment? 
When  its  vertex  is  in  the  arc  of  the  segment,  and  its  sides 

pass  through  the  ends  of  the  arc. 

4.  Make  a  segment,  and  inscribe  in  it  three  angles. 

5.  What  truths  follow  immediately  from  JVa  2  ? 

(1)  Inscribed  angles  that  intercept  equal  arcs  are  equal. 

(2)  All  angles  inscribed  in  the  same  segment  are  equal. 
For  they  all  intercept  the  same  arc.     Thus,  the  A  ACB, 

ADB,  AEB  (Fig.  108),  all  intercept  the  arc  AB. 

(3)  Every  angle  inscribed  in  a  semicircle  is  a  right  angle. 
For  the  intercepted  arc  is  a  semi-circumference. 

Thus,  the  A  ACB,  ADB,  AEB  (Fig.  109)  are  right  angles. 

D  n 


FIG.  108. 


O 
FIG.  109. 


6.  What  is  the  value  of  an  inscribed  angle  if  the  inter- 
cepted arc  is  60°?   153°?  320°?   118°  35'? 

7.  What  value  has  an  angle  inscribed  in  a  segment  whose 
arc  contains  60°?  90°?   135°?  180°?  270°?  300°? 

8.  Prove  the  construction  in  Fig.  27,  p.  19,  to  be  correct. 
Construct  a  right  triangle,  having  given  : 

9.  Hypotenuse  =  3  in.,  one  leg  =  If  in. 

10.  Hypotenuse  =  3  in.,  altitude  upon  hypotenuse  =  1  in. 

11.  Hypotenuse  =  3  in.,  one  acute  angle  =  60°. 

12.  A  point  A  is  2  miles  from  a  line  BC ;;    BC=3  miles, 
/  BAC=  90°.     Find  AB  and  AC  (by  drawing  to  scale) . 


102  LESSONS   IN   GEOMETRY. 


Lesson   53. 

1.  Define  a  tangent  to  a  circle. 

A  tangent  to  a  circle  is  a  straight  line  which  touches  the 
circumference  at  a  point,  but  does  not  cut  it. 

A  tangent  is  said  to  touch  the  circle.  The  point  where  it 
touches  the  circle  is  called  the  point  of  contact. 

2.  Theorem.—,^  perpendicular ;  erected  at  the  end 
of  a  radius  is  a  tangent  to  the  circle. 

HYPOTHESIS. 

A  circle  with  centre  0. 

A  radius  OA.  [  o 

The  line  MAN  JL  to  OA. 


CONCLUSION. 

The  line  MN  is  a  tangent  to     W 
the  circle  at  the  point  A. 

PROOF.     Join  0  to  any  other  point  in  MN,  as  B. 

Then  we  know  that  OA  <  OB  (No.  3,  p.  71). 

Therefore  B  must  lie  outside  the  circle.  In  the  same  way 
we  can  show  that  every  point  in  MN  except  A  must  lie 
outside  the  circle. 

Therefore,  by  definition,  the  line  MNia  a  tangent,  and  A 
is  the  point  of  contact. 

3.  Mark  a  point  A  in  a  line  MN,  and  then  describe  three 
circles  which  shall  touch  MN  at  the  point  A.     Take  for  the 
radii  of  the  circles  1  in.,  2  in.,  and  3  in.     How  many  circles 
touching  MN  at  A  can  be  described  ?     In  what  line  do  their 
centres  lie  ? 

4.  Draw  a  line  MN,  and  then  describe  three  circles  with 
the  same  radius,  all  of  which  shall  touch  MN.     How  many 
circles,  all  having  the  same  radius  and  all  touching  MN,  can 
be  described  ?    In  what  lines  do  their  centres  lie  ? 


THE   CIRCLE. 


103 


5.  Problem.  —  To  draw  a  tangent  to  a  given  circle 
through  a  given  point  A. 

Case  1  (Fig.  111).     Let  A  be  in  the  given  circumference. 

CONSTRUCTION.     Draw  the  radius  OA,  and  through  A  draw 
the  line  BAG  _L  to  OA.     BAG  is  the  required  tangent. 

PROOF.     See  No.  2. 
B 


FIG.  111. 


FIG.  112. 


Case  2  (Fig.  112).     Let  A  be  outside  the  given  circle. 

CONSTRUCTION.     Join  AO. 

LTpon  AO  as  diameter  describe  a  circle. 

This  circle  will  cut  the  given  circle  at  two  points,  B  and  G. 

Draw  the  lines  AB  and  AC. 

Then  AB  and  AC  are  tangents  to  the  given  circle. 

PROOF.     The  angles  ABO,  AGO,  are  right  angles.    Why? 
Also  B  and  G  are  the  ends  of  radii. 
Therefore  AB  and  AC  are  tangents.    Why? 

6.  Prove  that  the  tangents  AB,  AC  (Fig.  112),  are  equal. 

7.  Describe  a  circle  with  a  radius  of  2  in.,  and  draw  tan- 
gents to  it  from  a  point  3J  in.  from  the  centre. 

8.  Describe  a  circle  passing  through  a  given  point,  and 
touching  a  given  line  at  a  given  point  in  the  line. 

Describe  a  circle,  and  then  draw  a  tangent : 

9.  Parallel  to  a  given  straight  line. 

10.    Perpendicular  to  a  given  straight  line. 


104 


LESSONS   IN   GEOMETKY. 


Lesson  54. 

1.  When  are  two  circles  said  to  touch  each  other? 
Two  circles  are  said  to  touch  each  other,  or  to  be  tangent 

to  each  other,  when  they  both  touch  the  same  straight  line  at 
the  same  point. 

If  the  circles  lie  on  opposite  sides  of  the  straight  line,  they 
are  said  to  touch  each  other  externally  (Fig.  113). 

If  they  lie  on  the  same  side  of  the  straight  line,  they  are 
said  to  touch  each  other  internally  (Fig.  114). 

In  both  cases,  the  line  is  a  common  tangent  to  the  circles. 

2.  What  truth  easily  follows  from  No.  7  ? 

If  two  circles  touch  each  other,  the  radii  drawn  to  the 
point  of  contact  are  both  perpendicular  to  the  common  tan- 
gent (No.  2,  p.  102).  Therefore  they  must  lie  in  the  same 
straight  line.  In  other  words,  the  centres  and  the  point  of 
contact  lie  in  the  same  straight  line. 


FIG. 113. 


FIG.  114. 


3.  Construct  an  equilateral  triangle,  side  2  in.,  and  by 
means  of  it  describe  three  circles,  so  that  each  circle  shall 
touch  the  other  two  circles. 

4.  Describe  the  smallest  circle  which  can  touch  a  given 
circle  and  pass  through  a  given  point  exterior  to  the  given 
circle. 

5.  Describe  a  circle  touching  two  given  parallel  lines,  and 
passing  through  a  given  point  between  the  lines. 


THE  CIRCLE.  105 

6.  Draw  an  angle,  and  describe  three  different  circles, 
each  touching  the  sides  of  the  angle.     How  many  circles 
touching  the  sides  of  a  given  angle  can  be  described?     In 
what  line  do  their  centres  lie  ? 

7.  Draw  a  triangle,  and  describe  a  circle  having  its  centre 
in  one  side  and  touching  the  other  two  sides. 

8.  How  many  common  tangents  can  be  drawn  in  Fig.  113? 

9.  Define    circumscribed    figures    and    inscribed 
figures. 

A  circle  is  circumscribed  about  a  polygon  if  its  circum- 
ference passes  through  all  the  vertices  of  the  polygon ;  and 
the  polygon  is,  in  this  case,  inscribed  in  the  circle. 

A  circle  is  inscribed  in  a  polygon  if  it  touches  all  the  sides 
of  a  polygon  ;  and  in  this  case  the  polygon  is  circumscribed 
about  the  circle. 

10.  Illustrate  by  figures  the  definitions  in  No.  9. 

11.  Draw  a  triangle,  and  then  circumscribe  a  circle  about 
the  triangle  (see  p.  99,  No.  5). 

12.  Inscribe  a  circle  in  a  given  triangle.     (Bisect  any  two 
angles.     Prove  your  construction  correct.    See  No.  6,  p.  73.) 

13.  Circumscribe  a  circle  about  an  equilateral  triangle, 
and  also  inscribe  a  circle  in  the  same  triangle. 

14.  Inscribe  a  circle  in  a  square  whose  side  is  2  in. 

15.  Construct,,  a  square  of  side  3  in.,  inscribe  a  circle  in 
the  square,  and  form  an  octagon  by  cutting  off  the  corners 
with  tangents  to  the  circle. 

16.  Construct  a  square  of  side  4  in.,  and  then  describe 
four  equal  circles,  so  that  each  circle  shall  touch  two  of  the 
other  circles,  and  also  two  sides  of  the  square. 

17.  Draw,  a  rhombus,  and  inscribe  in  it  a  circle. 


106 


LESSONS   IN   GEOMETRY. 


Lesson  55. 

1.  Define  the  centre  of  a  regular  polygon. 

The  centre  of  a  regular  polygon  is  a  point  equidistant  from 
all  the  vertices  of  the  polygon. 

2.  Theorem.  —  The  point  where  the  bisectors  of  any 
two  angles  of  a  regular  polygon  meet,  is  the  centre 
of  the  polygon. 

HYPOTHESIS. 

Let  ABCDE  be  a  regular  polygon. 
Let  the  bisectors  of  the  A  A  and  B 
meet  at  0. 

CONCLUSION.  AO,  BO,  CO,  DO,  EO, 
are  all  equal. 

PROOF.     Successive  steps.      Supply 
all  the  reasons. 

(1)  A  OAB  is  isosceles,  and  OA=  OB. 

(2)  A  OBC  =  A  OAB  (No.  14,  p.  66). 

(3)  A  OCD  =  A  OBC  (No.  14,  p.  66). 

(4)  A  ODE=  A  OCD  (No.  14,  p.  66). 


FIG.  115. 

What  follows? 
What  follows? 
What  follows  ? 


3.  What  truths  follow  immediately  from  No.  2  ? 

(1)  The  lines  draivn  from  the  centre  of  a  regular  polygon 
to  the  vertices  bisect  all  the  angles. 

(2)  A   circle   can   be   circumscribed   about   every  regular 
polygon. 

4,  Theorem.  —  The  centre  of  a  regular  polygon  is 
equidistant  from  all  the  sides. 

What  is  the  hypothesis  (use  Fig.  115)  ? 
What  is  the  conclusion  ? 

PROOF.     Compare  the  right  triangles   OFB,   0GB ;    and 
then  OGC,  OHC,  etc.  (see  p.  63,  No.  3;  p.  67,  No.  2). 


THE   CIRCLE. 


107 


5.  What  truths  follow  immediately  from  JV0.  4  ? 

(1)  Perpendiculars  dropped  from  the  centre  of  a  regular 
polygon  to  the  sides  bisect  the  sides. 

(2)  A  circle  can  be  inscribed  in  every  regular  polygon. 

6.  Define  the  radius  and  the  apothem  of  a  regular 
polygon. 

The  radius  of  a  regular  polygon  is  a  line  drawn  from  the 
centre  to  any  one  of  the  vertices. 

The  apothem  of  a  regular  polygon  is  a  perpendicular 
dropped  from  the  centre  to  any  one  of  the  sides. 

7.  Show  how  to  find  the  centre  of  a  regular  polygon. 

8.  Inscribe  a  square  in  a  given  circle. 

To  do  this,  draw  two  perpendicular  diameters,  and  join 
their  ends,  taken  in  order  (Fig.  116).  Prove  that  the  figure 
thus  obtained  is  a  square. 

9.  Inscribe  in  a  given  circle  a  regular  octagon. 


FIG.  117. 

10.  Inscribe  in  a  given  circle  a  regular  hexagon. 

To  do  this,  apply  the  radius  as  a  chord  six  times  (Fig.  117). 
Then  prove  that  the  figure  thus  formed  is  a  regular  hexagon 
(show  that  Z  AOB  =  60°) . 

11.  Inscribe  in  a  given  circle  an  equilateral  triangle. 

12.  Inscribe  in  a  given  circle  a  regular  dodecagon. 

13.  Construct  a  regular  hexagon  with  a  side  equal  to  3  in. 


108  LESSONS   IN   GEOMETRY. 


Lesson  56. 

1.  What  is  the  relation  between  the  circumference 
of  a  circle  and  the  diameter  ? 

The  diameter  of  a  circle  is  contained  in  the  circumference  a 
little  more  than  three  times.  The  exact  value  of  the  quotient 
cannot  be  expressed  by  a  number,  but  it  is  known  to  be  the 
same  for  all  circles,  and,  for  convenience,  is  represented  by 
the  Greek  letter  TT  (pronounced  like  pie)  . 

The  value  of  TT,  accurate  enough  for  most  purposes,  is 

3*  or  f  - 

The  value  of  ?r,  correct  to  five  decimal  places,  is  3.14159. 
If  we  take  TT  =  ^2-,  the  relation  between  circumference  and 
diameter  is  expressed  by  either  of  the  following  equations  : 

Circumference  =  diameter  x  3?-. 
Diameter         =  circumference  x  -^V 

The  relation  is  often  expressed  in  the  following  general 
form,  in  which  c  denotes  circumference,  and  r  radius  : 


NOTE.  —  The  method  by  which  the  value  of  TT  is  shown  to  be  the 
same  for  all  circles,  and  the  process  of  computing  its  approximate 
values,  require  the  aid  of  theorems  which  are  not  given  in  this  book. 

The  value  has  been  computed  to  over  700  decimal  places. 

Find  the  circumference  of  a  circle,  having  given  : 

2.  Radius  7  in.  6.    Diameter  77  ft. 

3.  Radius  21  in.  7.    Diameter  49  ft. 

4.  Radius  40  ft.  10  in.  8.    Diameter  18  yds.  2  ft. 

5.  Radius  6  yds.  1  ft.  3  in.         9.    Diameter  2T\  in. 

10.  The  circumferences  of  two  concentric  circles  are  16J 
feet  and  18£  ft.  Find  the  width  of  the  ring  between  them. 

w 


THE   CIRCLE.  109 

11.  If  the  radius  of  a  circle  is  3^  in.,  what  is  the  length  of 
an  arc  of  30°  ? 

The  circumference  =  2  X  f  X  ty  in.  =  22  in. 
The  circumference  is  divided  into  360°. 
Therefore  an  arc  of  I0  =  ^j-  in. 

And  an  arc  of  30°  =  22x8°  in.  =  lf  in. 

Find  the  length  of  an  arc,  having  given  : 

12.  Circumference  100  in.,  angle  at  centre  30°. 

13.  Radius  14  in.,  angle  at  centre  45°- 

14.  Radius  42  ft.,  angle  at  centre  11°  15'. 

15.  Diameter  70  in.,  angle  at  centre  36°. 
Find  the  radius  of  a  circle,  having  given  : 

16.  Circumference  132  ft. 

17.  Circumference  198  ft. 

18.  Arc  of  60°=  G6  ft. 

19.  Arc  of  15°=  3  ft.  8  in. 

20.  A  wheel  makes  220  revolutions  in  going  half  a  mile. 
What  is  its  diameter  ? 

21.  A  wheel  whose  diameter  is  3J  ft.  made  1200  revolu- 
tions in  going  a  certain  distance.     Find  the  distance. 

22.  How  deep  is  a  well  if  the  wheel,  whose  diameter  is 
2  ft.  4  in.,  makes  30  revolutions  in  raising  the  bucket? 

23.  What  angle  at  the  centre  will  intercept  an  arc  of  6  ft. 
5  in.  if  the  radius  of  the  circle  is  8  ft.  2  in.  ? 

Circumference  —  2  X  -27-  X  98  in.  —  616  in. 


Since  616  in.  contain  360°,  1  in.  of  arc  will  contain       -,  and  77  in. 

will  contain77*3600,  or  45°. 
616 

24.    If  the  radius  of  a  circle  is  7  in.,  what  angle  at  the 
centre  will  intercept  an  arc  7  in.  long  ? 


110  LESSONS   IN   GEOMETRY. 

Lesson  57.     Review. 

1.  Review  all  the  italicized  exercises  in  Lessons  50-53. 

2.  Make  an  arc  of  225°. 

3.  Make  an  arc  equal  to  twice  a  given  arc. 

4.  Make  a  sector  equal  to  twice  a  given  sector. 

5.  Describe  a  circle  with  a  radius  of  2  in.  and  passing 
through  two  given  points  A  and  _B.     When  is  the  solution  of 
the  problem  impossible? 

6.  If  airy  number  of  parallel  chords  are  drawn  in  a  circle, 
upon  what  line  are  all  their  middle  points  located  ? 

7.  Through  a  given  point  within  a  circle  draw  that  chord 
which  is  bisected  at  the  given  point. 

Prove  that  your  construction  is  correct. 

8.  A  segment  is  made  by  joining  the  ends  of  the  arc  of 
a  quadrant.    What  is  the  value  of  any  angle  inscribed  in  this 
segment? 

9.  If  a  series  of  circles  are  made,  all  of  them  touching 
two  parallel  lines,  upon  what  line  will  all  their  centres  lie? 

10.  Describe  a  circle  touching  two  given  parallel  lines, 
one  of  them  at  a  given  point  A. 

11.  A  man  wishes  to  locate  his  house  so  that  it  shall  be 
2  miles  from  a  church,  and  equidistant  from  the  homes  of 
two  of  his  friends.     Show  by  a  construction  how  the  proper 
location  may  be  found. 

12.  A  man  walks  3  miles  in  a  straight  line  ;  and  then  walks 
in  the  arc  of  a  circle  through  whose  centre  he  has  passed, 
and  whose  radius  is  1  mile,  till  he  has  made  the  chord  of  the 
arc  equal  to  the  radius.     Find  how  far  he  is  from  home. 


THE  CIRCLE.  Ill 

Lesson  58.    Review. 

1.  Review  all  the  italicized  exercises  in  Lessons  54-56. 

2.  Describe  a  circle,  and  upon  one  of  its  radii  OA  as  a 
diameter   describe    another   circle.      Through   A  draw  any 
straight  line,  meeting  the  circumferences  again  at  B  and  C. 
Prove  that  AB  =  BC. 

3.  Make  two  equal  circles  touching  each  other  externally, 
and  draw  all  the  common  tangents. 

4.  Make  two  equal  circles,  exterior  to  each  other  and 
not  touching.     Then  draw  all  the  common  tangents. 

5.  Inscribe  a  regular  hexagon  in  a  circle,  and  then  cir- 
cumscribe a  regular  hexagon  about  the  same  circle.     This  is 
done  by  drawing  tangents  through  the  vertices  of  the  in- 
scribed hexagon.     Can  you  prove  that  the  figure  formed  by 
these  tangents  is  a  regular  hexagon  ? 

6.  Inscribe  four  equal  circles  in  a  given  square  so  that 
each  circle  shall  touch  two  other  circles  and  one  side  only  of 
the  square. 

7.  A  circular  park  is  laid  out  having  a  radius  of  350  ft. 
What  will  it  cost  to  build  a  stone  wall  around  the  park  at 
$8.00  per  yard? 

8.  The  latitude  of  Leipsic  is  51°  21',  that  of  Venice  is 
45°  26'.     Venice  is  due  south  of  Leipsic.     How  many  miles 
are  they  apart?    (Take  as  the  radius  of  the  earth  4000  miles.) 

9.  If  the  radius  of  a  circle  is  4  ft.  8  in.,  what  is  the 
perimeter  of  a  sector  whose  angle  is  45°? 

10.  Draw  four  intersecting  lines  so  as  to  make  four  tri- 
angles, and  circumscribe  a  circle  about  each  of  these  triangles. 
If  your  work  is  correct,  the  four  circumferences  will  be  found 
to  pass  through  the  same  point. 


112  LESSONS   IN   GEOMETRY. 

Lesson  59.    Review. 

NOTE.  —  In  the  exercises  of  this  lesson  take  7r  =  3.1416. 

1.  Review  all  the  italicized  exercises  in  Chapter  VI. 

2.  Find  the  circumference  of  a  circle  if  the  radius  =  10m. 

3.  Find  the  radius  of  a  circle  if  the  circumference  =  10m. 

4.  What  must  be  the  diameter  of  a  round  dining-table 
for  12  persons,  if  60cm  is  allowed  to  each  person? 

5.  A  wheel  75cm  in  diameter  makes  3000  revolutions  in 
going  a  certain  distance.     What  is  the  distance  ? 

6.  How  many  trees  10m  apart  can  be  set  out  around  the 
edge  of  a  circular  pond  500m  in  diameter  ? 

7.  The  radii  of  two  concentric  circumferences  are  8cm 
and  10cm.     Find  the  length  of  the  circumference  situated  just 
half  way  between  them. 

8.  Two  toothed  wheels  work  together.    One  has  24  teeth, 
the  other  has  144  teeth.     The  distance  between  the  centres 
of  the  teeth  is  the  same  on  each  wheel.     How  many  revolu- 
tions will  the  second  wheel  make  while  the  first  is  making 
600?     If  the  radius  of  the  first  wheel  is  4cm,  what  is  the 
radius  of  the  other? 

9.  Describe  a  circle  with  a  radius  of  2cm,  and  then  cir- 
cumscribe about  the  circle  a  square.     Find  (to  the  nearest 
millimeter)  the  difference  in  length  between  the  circumference 
of  the  circle  and  the  perimeter  of  the  square. 

Diameter  of  a  circle  =  10cm.     Find  the  length  of  the  arc : 

10.  Intercepted  by  a  side  of  the  inscribed  square. 

1 1 .  Intercepted  by  a  side  of  the  inscribed  regular  pentagon. 

12.  Intercepted  by  a  side  of  the  inscribed  regular  hexagon. 

13.  Intercepted  by  a  side  of  the  inscribed  regular  octagon. 


.     CHAPTER   VII. 


AREAS. 

Lesson  6O. 

1.  How  are  surfaces  measured? 

Surfaces,  like  lines,  are  measured  by  choosing  a  unit,  and 
then  finding  how  many  times  this  unit  is  contained  in  the 
surface  which  we  wish  to  measure. 

The  number  of  times  the  unit  is  contained  in  the  surface 
to  be  measured,  followed  by  the  name  of  the  unit,  is  called 
the  area  of  the  surface. 

2.  What  units  of  area  are  in  common  use  ? 

The  square  inch,  the  square  foot,  the  square  yard,  the  acre, 
and  the  square  mile.  All  these  units,  except  the  acre,  are 
squares  whose  sides  are  equal  to  the  units  of  length. 

Their  abbreviations  are  formed  by  prefixing  to  those  of 
the  corresponding  units  of  length  the  letters  "  sq."  (standing 
for  "  square"). 

3.  How  are  those  units  related  ? 

These  units  are  related  as  shown  in  the  following  table : 


144  sq.  in.  =  1  sq.  ft. 
9  sq.  ft.  =  1  sq.  yd. 


43,560  sq.  ft.=  l  acre. 
640  acres  =  1  sq.  mile 


4.  How  many  square  feet  make  1  square  mile? 

5.  How  many  square  feet  in  J  of  an  acre? 

6.  Reduce  16  sq.  in.  to  the  fraction  of  a  square  foot. 

7.  Reduce  -|  of  a  square  foot  to  square  inches. 

8.  A  man  bought  an  acre  of  land  for  $5000,  and  sold  it 
for  20  cents  per  square  foot.    How  much  money  did  he  make  ? 


114 


LESSONS  IN   GEOMETRY. 


FIG.  118. 


9.  How    can    a    square    ~be    divided    into   smaller 
squares  ? 

Let  ABCD  (Fig.  118)  be  a  square. 

Divide  AB  into  any  number  of  equal 
parts  (here,  six). 

Draw  through  the  points  of  division 
lines  parallel  to  AD. 

These  lines  will  divide  the  square 
into  six  equal  rectangles. 

Divide  also  AD  into  six  equal  parts, 
and  draw  through  the  points  of  division  lines  parallel  to  AB. 
These  lines  will  subdivide  each  rectangle  into  six  equal 
squares.  Therefore  the  entire  square  is  now  divided  into 
6x6  or  36  equal  squares. 

In  general,  the  number  of  equal  squares  into  which  a  square 
is  decomposed  by  this  process  is  found  by  multiplying  the 
number  of  equal  parts  in  one  of  the  sides  by  itself. 

10.  Problem.  —  To  find  the  area  of  a  square. 

Let  ABCD  (Fig.  119)  be  a  square. 

Measure  one  side  AB. 

Let,  for  example,  AB  =  4  in. 

By  proceeding  as  in  No.  9,  we  see 
that  the  area  of  the  square  will  be 
4x4  or  16  sq.  in. 

Next,  let  JJ3  =  4.25  in.  The  same  A 
reasoning  will  still  apply,  only  we  shall 
now  have  in  AB  425  equal  parts,  each  a  hundredth  of  an 
inch  long,  and  in  ABCD,  425x425  or  180,625  equal 
squares.  Now  one  inch  contains  100  of  these  equal  parts. 
Therefore^  one  square  inch  will  contain  100  x  100  or  10,000 
of  the  equal  squares.  Therefore  the  area  of  the  square 
ABCD,  in  square  inches,  will  be  \Vimr  or  18.0625,  a 
result  at  once  found  by  multiplying  4.25  by  itself. 


AREAS.  115 

In  general,  by  multiplying  by  itself  the  number  of  units  of 
length  in  a  line  AB,  we  obtain  the  number  of  units  of  area 
in  the  square  constructed  upon  AB  as  a  side  ;  the  unit  of 
area  being  always  the  square  whose  side  is  equal  to  the  unit 
of  length. 

Multiplying  a  number  by  itself  is  called  squaring  it,  and 
the  square  of  the  number  of  units  of  length  in  a  line  AB  is 
called,  for  brevity,  "  the  square  of  AB,"  and  written  Aff. 
Hence  the  result  may  be  briefly  expressed  as  follows : 

Area  of  a  square  =  square  of  one  side. 

Find  the  area  of  a  square  if  one  side  is  equal  to : 

11.  9  in.  16.    1  ft.  8  in. 

12.  14  ft.  17.    2  ft.  4  in. 

13.  18  yds.  18.    4  yds.  1  ft.  2  in. 

14.  1\  in  19.    98  ft.  6  in. 

15.  2i  in.  20.    56  ft.  4  in. 

21.  Find  the  area  in  acres  and  square  feet  of  a  square 
field  each  side  of  which  is  178  yds.  long. 

22.  How  many  tiles  8  in.  square  are  required  to  cover  a 
square  court  one  side  of  which  is  33  ft.  4  in.  long? 

23.  What  will  it  cost,  at  20  cents  per  square  foot,  to  paint 
a  square  floor  the  side  of  which  measures  18J  ft. 

24.  Find  the  side  of  a  square  whose  area  is  2000  sq.  ft. 

SOLUTION.  —We  must  take  the  square  root  of  2000.  V2000=44.72  +. 
Hence  the  side  (correct  to  inches)  =  44.72  ft.  — 44ft.  9  in. 

25.  Find  the  side  of  a  square  whose  area  is  2J  sq.  yds. 

26.  Find  the  side  of  a  square  whose  area  is  6000  sq.  ft. 

27.  What  will  it  cost,  at  $8  per  yard,  to  enclose  with  an 
iron  fence  a  square  park  containing  1000  acres? 

23.    Explain  why  it  requires  144  sq.  in.  to  make  1  sq.  ft. 


116 


LESSONS   IN   GEOMETRY. 


Lesson  61. 

1.  Problem.  —  To  find  the  area  of  a  rectangle. 
Let  ABCD  (Fig.   120)  be  a   p 


FIG.  120. 


B 


rectangle. 

Measure  AB  and  AD. 

Let  AB  =  7  in. ,  AD  =  4  in. 

We  can  show  by  reasoning 
as  in  No.  9,  p.  114,  that  the 
area  of  the  rectangle  is  equal 
to  7  X  4  or  28  sq.  in. 

In  general,  the  number  of  units  of  area  in  a  rectangle  is 
found  by  multiplying  the  number  of  units  of  length  in  one 
side  by  the  number  of  units  of  length  in  the  adjacent  side. 

The  adjacent  sides  are  equal  to  the  dimensions  of  the  rec- 
tangle, and  are  usually  called  length  and  breadth ;  hence  we 
have  the  following  condensed  formula  : 

Area  of  a  rectangle  =  length  x  breadth. 

Find  the  area  of  a  rectangle,  having  given : 

2.  Length  12  in.,  breadth  9  in. 

3.  Length  15 J  in.,  breadth  6  in. 

4.  Length  3^  in.,  breadth  2^  in. 

5.  Length  200  ft.,  breadth  60  ft. 

6.  Length  5  ft.  6  in.,  breadth  1  ft.  6  in. 

7.  Length  4  yds.  1  ft.  6  in.,  breadth  6  ft. 

Find  the  other  dimension  of  a  rectangle,  having  given : 

8.  Area  100  sq.  in.,  one  dimension  8  in. 

9.  Area  288  sq.  in.,  one  dimension  6  in. 

10.  Area  22^  sq.  in.,  one  dimension  2£  in. 

11.  Area  180  sq.  ft.,  one  dimension  4  yds. 


AREAS.  117 

12.  How  many  bricks  9  in.  by  4  in.  are  required  to  cover 
a  floor  34  ft.  long  and  17  ft.  wide? 

13.  A  street  ^  a  mile  long  has  on  each  side  a  sidewalk  7J 
ft.  wide.     Find  the  cost  of  paving  the  sidewalk  with  stones 
2  ft.  9  in.  by  1  ft.  8  in.,  and  costing,  laid  down,  $1  each. 

14.  A  lawn  measures  144  yds.  by  98  yds.  ;  the  turfs  which 
are  used  measure  18  in.  by  14  in.,  and  cost,  laid  down,  75 
cents  per  dozen.     Find  the  cost  of  turfing  the  lawn. 

15.  A  floor  is  27  ft.  6  in.  long  and  8  yds.  wide.     If  the 
planks  to  cover  it  measure  11  ft.  by  9  in.,  how  many  are 
needed,   and  what  will   they  cost  at  10  cents    per  square 
foot? 

16.  How  many  yards  of  paper  27  in.  wide  are  required  to 
paper  a  room  18  ft.  long,  12  ft.  wide,  and  11  ft.  high? 

17.  What  will  it  cost  to  line  with  lead,  at  2  cents  per 
square  inch,  the  inside  of  an  open  cistern  10  ft.  long,  6  ft. 
wide,  and  4  ft.  deep? 

18.  A  man  wants  4  sq.  ft.  of  wood,  and  has  only  a  plank 
18  in.  wide  from  which  to  cut  it  off.     Find  the  length  of  the 
piece  he  must  cut  off. 

19.  The  dimensions  of  a  rectangle  are  36  ft.  and  20  ft. 
If  the  length  be  diminished  by  6  ft. ,  how  much  must  be  added 
to  the  breadth  in  order  that  the  area  may  remain  the  same  ? 

20.  A  rectangular  field,   half  a  mile  long,  contains  100 
acres.     Find  the  width  of  the  field  in  feet. 

21.  A  room  is  39  ft.  by  23  ft.  and  17  ft.  high,  with  three 
windows  each  9^  ft.  by  8  ft. ;  two  doors  each  10J  ft.  by  6  ft. ; 
and  two  fireplaces  each  6£  ft.  by  4  ft.     The  carpet  is  2  ft. 
wide,  and  costs  $1.50  per  yd.  ;  the  paper  is  2  ft.  wide,  and 
costs  75  cents  per  roll  of  10  yds.     Find  the  cost  of  carpet- 
ing and  papering  this  room,  the  carpet  running  lengthwise, 
and  the  labor  costing  $12, 


LESSONS   IN   GEOMETRY. 

* 

Lesson  62. 

1.  Problem.  —  To  find  the  area  of  a  parallelogram. 

Let  ABCD  (Fig.  121)  be  a  parallelogram,  having  the  base. 
AB  and  the  altitude  BE.          F  D  EC 

Draw  AF  _L  to  AB,  and 
meeting  AB  produced  at  the 
point  F. 

The  figure  ABEF  is  a 
rectangle  having  the  same  FlG- 12L 

base  AB  and  altitude  BE  as  the  parallelogram. 

ABCE  =  AADF.     Why? 

Entire  figure    ABCF-A  ADF  =  parallelogram  ABCD. 
Entire  figure     ABCF—A  BCE  =  rectangle  ABEF. 
Therefore,  parallelogram  ABCD  =  rectangle  ABEF. 
But  rectangle  ABEF=  ABxBE(p.ll6,No.l). 

Therefore,  parallelogram  ABCD  =  AB  x  BE. 

Hence,  the  area  of  a  parallelogram  is  found  by  measuring 
its  dimensions  and  multiplying  their  values  together. 

Area  of  a  parallelogram  =  base  x  altitude. 

2.  Does  this  formula  also  hold  true  for  the  square  and  for 
the  rectangle  ? 

Find  the  area  of  a  parallelogram,  having  given : 

3.  Base  16  in.,  altitude  5  in. 

4.  Base  2J  in.,  altitude  14  in. 

5.  Base  1  ft.  9  in.,  altitude  4  ft. 

6.  Base  15  ft.,  altitude  equal  to  half  the  base. 

7.  The  base  of  a  parallelogram  =  18  in.     What  must  be 
its  altitude  in  order  that  it  may  contain  exactly  1  sq.  ft.  ? 


AREAS.  119 

8.  What  conclusion  may  be  drawn  from  JV0.  1  re- 
specting all  parallelograms  that  have  equal  bases 
and  equal  altitudes? 

Parallelograms  having  equal  bases  and  equal  altitudes  have 
also  equal  areas;  in  other  words,  they  are  equivalent. 

For  by  No.  1  each  one  of  the  parallelograms  is  equivalent 

to  a  rectangle  having  an  H      N  •         G  F  M     D       E C 

equal  base  and  an  equal 
altitude.  For  example 
(Fig.  122)  :  the  parallelo- 
grams ABCD,  ABEF, 
and  ABGH  are  equiva- 
lent, each  being  equivalent  to  the  rectangle  ABMN. 
9.  Construct  a  set  of  four  equivalent  parallelograms. 

10.  Make   two    parallelograms,    differing   very   much   in 
shape,  but  having  the  same  area. 

11.  Construct  a  parallelogram,  and  then  find  its  area. 

12.  The  bases  of  two  parallelograms  are  each  9|  in.  long. 
The  altitude  of  one  is  1  in.,  and  the  altitude  of  the  other 
is  3  in.     How  much  greater  is  the  area  of  one  than  the  area 
of  the  other? 

13.  The  bases  of  two  parallelograms  are  equal ;  the  alti- 
tude of  one  is  three  times  that  of  the  other.     Compare  their 
areas.     Draw  a  figure  to  illustrate  your  answer. 

14.  The  altitudes  of  two  parallelograms  are  equal ;    the 
base  of  one  is  three  times  that  of  the  other.     Compare  their 
areas.     Draw  a  figure  to  illustrate  your  answer. 

15.  Find  the  area  of  a  rhombus,  the  perimeter  of  which  is 
156  ft.,  and  the  altitude  9  ft.  4  in. 

16.  The  area  of  a  field  having  the  shape  of  a  rhombus  is 
exactly  10  acres.     The  distance  from  any  one  side  to  the 
opposite  side  is  400  ft.     Required  the  cost  of  fencing  the 
field  at  $2.50  per  rod. 


120 


LESSONS   IN  GEOMETRY. 


Lesson  63. 

1,  Problem.  —  To  find  the  area  of  a  triangle. 

Let  ABC  (Fig.  123)  be  a  triangle,  AB  the  base,  CD  the 
altitude. 

Draw  AE  \\  to  BC,  and  CE  II  to  AB ;  the  figure  ABCE  is 
a  parallelogram  having  AB  for  base,  CD  for  altitude. 

Therefore,  A  ABC  =  A  AEC   (p.  84,  No.  3) . 

Hence,    area  of  A  ABC  —  %  area  of  parallelogram  ABCD. 
Now,       area  of  parallelogram  ABCD  =  AB  x  CD. 
Hence,    area  of  A  ABC  =  %AB  x  CD  ;  or, 

Area  of  a  triangle  =  %  x  base  x  altitude. 


FIG.  123. 

2.  What  follows  immediately  from  this  formula? 
A  having  equal  bases  and  equal  altitudes  are  equivalent. 
Thus  (Fig.  124)  the  A  ABC,  ABD,  ABE,  are  equivalent. 
Find  the  area  of  a  triangle,  having  given : 

3.  Base  11  in.,  altitude  10  in. 

4.  Base  2^  in.,  altitude  1  in. 

5.  Base  3  ft.  6  in.,  altitude  2  ft. 

6.  Base  20  ft.  6  in.,  altitude  10  ft.  4  in. 

7.  Find  the  area  of  a  right  triangle  if  the  lengths  of  the 
two  legs  are  4J  in.  and  3^  in. 

8.  A  garden  is  laid  out  in  the  shape  of  a  right  triangle. 
One  leg  =  150  yds.,  the  other  leg  =  200  yds.     Find  the  value 
of  the  garden  at  12  J  cents  per  square  foot. 


AKEAS. 


121 


9.  Apply  the  formula  of  No.  1  to  a  rhombus. 

Let  ABCD  (Fig.  125)  be  a  rhombus.  The  diagonals  AC 
and  BD  bisect  each  other  at  right  angles  (p.  86,  No.  3). 
AC  divides  the  rhombus  into  two  equal  isosceles  triangles, 
each  having  AC  for  base,  and  half  of  BD  for  altitude. 

The  area  of  each  A  =  \AC  x  %BD  ;  therefore  the  area  of 
the  rhombus  =  \  X  AC  X  BD  ;  or,  in  general : 

Area  of  a  rhombus  =  half  the  product  of  the  diagonals. 


E 

FIG.  126. 


B 


10.  Apply  the  formula  of  No.  I  to  a  trapezoid. 

Let  ABCD  (Fig.  126)  be  a  trapezoid,  AB  and  CD  the 
bases,  DE  the  altitude.  Draw  the  diagonal  DJ5,  dividing 
the  trapezoid  into  the  two  triangles  ABD  and  BDC. 

Area  of  A  ABD  =  %ABx  DE.     Why  ? 

Area  of  A  BDC  =  \DC  x  DE. 

Hence  area  of  trapezoid  ABCD  =  £  (AB  +  DC)  X  DE. 

Putting  this  result  into  general  terms,  we  have  the  formula  : 
Area  of  a  trapezoid  =  J  (sum  of  bases)  x  altitude. 

Find  the  area  of  a  rhombus,  having  given  : 

11.  The  diagonals  9|  in.  and  6J  in. 

12.  The  diagonals  26  yds.  and  34|  yds. 

13.  Find  the  area  of  a  trapezoid  if  the  two  bases  are  37  ft. 
and  25  ft.,  and  the  altitude  19  ft. 

14.  Find  in  square  feet  the  area  of  ABCD  (Fig.  126)  if 
AB  =  i  mile,  CD  =  J  mile,  DE  =  1  mile. 


122 


LESSONS   IN  GEOMETRY. 


Lesson   64. 

1.  Problem.  —  To  find  the  area  of  a  regular  polygon. 

Let  ABCDE  (Fig.  127)  be  a  regular  polygon,  0  the 
centre,  OF  the  apothem. 

Draw  the  radii  OA,  OB,  etc. 

The   radii    divide  the   polygon   into 
the  equal  isosceles  A  AOB,  BOG,  etc. 

Altitude  of  each  A  =  OF. 

Area  of  A  AOB  =  ±ABx  OF. 

Area  of  A  BOG  =  %BC  x  OF,  etc. 

Adding  these  areas,  we  obtain  :  FI»-  127. 

Area  of  polygon  ABCDE  =  %(AB+BC+etc.)  x  OF',  or, 
Area  of  a  regular  polygon  =  j-  x  perimeter  x  apothem. 

If  one  side  of  a  regular  polygon  and  the  number  of  sides 
are  given,  the  apothem  may  be  found  (correct  enough  for  all 
practical  purposes)  by  multiplying  the  given  value  of  the  side 
by  the  proper  number  in  the  following  table  : 


No.  of  sides.  Apothem. 

3 one  side  X  0.288 

4 one  side  X  0.500 

5 one  side X0.688 

6 one  side  X  0.866 


No.  of  sides.  Apolhem. 

7  ......  one  side  X 1-038 

8 one  side  X 1-207 

10 one  side  X  1.539 

12  .  .  one  side  X  1.866 


Find  the  area  of: 

2.  An  equilateral  triangle,  if  one  side  =  5  in. 

3.  A  regular  pentagon,        if  one  side  =  6  in. 

4.  A  regular  octagon,          if  one  side  =15  ft. 

5.  A  regular  decagon,         if  one  side  =  19  ft. 

6.  A  park  has  the  shape  of  a  regular  hexagon  ;  each  side 
is  1000  ft.  long.     Find  the  value  of  the  park,  at  8  cents  per 
square  foot. 


AREAS. 


123 


7.  How  may  the  area  of  any  polygon  be  found  ? 
The  area  of  any  polygon  may  be  found  by  dividing  it  into 

triangles,  or  into'  triangles  and   trapezoids,   computing  the 
areas  of  these  parts,  and  then  adding  the  results. 

8.  Find  the  area  of   the  polygon   ABCD   (Fig.    128), 
having  given  :  AC,  400  yds.  ;  BE,  120  yds.  ;  DF,  80  yds. 


FIG.  128. 


FIG.  129. 


9.  Find  the  area  of  the  polygon  ABODE  (Fig.  129), 
having  given  :  BE,  108  yds.  ;  EC,  96  yds.  ;  AF,  49  yds.  ; 
HD,  35  yds. ;  CG,  67  yds. 

10.  Find  the  area  of  the  polygon  ABCDE  (Fig.  130), 
having  given  :  AC,  475  yds.  ;  BG,  160  yds.  ;  AF,  175  yds.  ; 
AG,  320  yds.  ;  AH,  420  yds.  ;  EF,  160  yds.  ;  DH,  90  yds. 


11.  Find  the  area  of  the  polygon  ABCDEF  (Fig.  131), 
having  given  :  AC,  300  yds.  ;  BK,  115  yds.;  FG,  63  yds.  ; 
EG,  54  yds. ;  FH,  283  yds. ;  DH,  60  yds.  ;  AF,  125  yds.  ; 
CF,  325  yds.  Z  AFC=  90°. 


124 


LESSONS   IN   GEOMETRY. 


Lesson  65. 

1.  Problem.  —  To  find  the  area  of  a  circle. 
Circumscribe  about  a  circle  a  regular  polygon  r  then 

Area  of  the  polygon  =  J  X  perimeter  X  apothem  ; 
or,  since  the  apothem  is  equal  to  the  radius  of  the  circle, 
Area  of  the  polygon  =  -J-  x  perimeter  x  radius. 

The  greater  the  number  of  sides  of  the  polygon,  the  nearer 
will  its  perimeter  approach  to  the  circumference  of  the  circle, 
and  its  area  to  the  area  of  the  circle. v 

If  the  polygon  had  1000  sides,  it 
would  not  differ  sensibly  from  the 
circle ;  if  it  had  10,000,000  sides,  the 
difference  would  of  course  be  still  less. 

But  however  many  sides  the  poly- 
gon may  have,  its  area  may  always  be 
found  by  the  formula  above  given  ;  and 
it  can  be  proved  that  this  formula  will 
still  hold  true  if  we  substitute  the  area 
of  the  circle  for  the  area  of  the  polygon,  and  the  circumfer- 
ence of  the  circle  for  the  perimeter  of  the  polygon.  Making 
these  substitutions,  the  formula  becomes  : 

Area  of  a  circle  =  £  X  circumference  x  radius. 

To  change  this  formula  into  a  more  practical  form,  we 
must  substitute  for  circumference  the  value  which  we  have 
previously  found  (see  p.  108)  ;  namely,  -^  X  diameter,  or 
4^  X  radius.  After  reducing  to  the  simplest  form,  we  have  : 

Area  of  a  circle  =  -2^  X  (radius)2 ; 
or,  if  one  prefers  to  use  the  symbols  TT  and  r, 
Area  of  a  circle  =^1^. 


AREAS.  125 

2.    From  the  formula  for  the  area  of  a  circle  find  the 
value  of  the  radius  in  terms  of  the  area. 
Take  the  equation  given  in  No.  1 : 

%f-  X  (radius)2  =  area. 

Multiply  both  sides  by  -^ ,  and  extract  the  square  root. 

Indicating  by  the  radial  sign  y'  the  square  root  of  the 
second  side,  we  obtain  for  the  result : 


radius  =     ^-  X  area. 
Find  the  area  of  a  circle,  having  given : 

3.  Radius  7  in. 

4.  Radius  14  ft. 

5.  Radius  40  ft.  10  in. 

6.  Radius  6  yds.  1  ft.  3  in. 

7.  Diameter  49  ft. 

8.  Diameter  18 1  yds. 

9.  Diameter  half  a  mile.     (Find  the  answer  in  acres.) 

10.  Find  the  radius,  if  the  area  =  24G4  sq.  ft. 

11.  Find  the  radius,  if  the  area=  6  acres. 

12.  Find  the  diameter,  if  the  area  =  1  sq.  mile. 

13.  The  radii  of  two  concentric  circles  are  3^  in.  and  7  in. 
Find  the  area  of  the  ring  bounded  by  their  circumferences. 

14.  Find  the  area  of  a  circular  ring  4  ft.  wide,  if  the 
radius  of  the  larger  circle  is  32  ft. 

15.  Out  of  a  square  piece  of  wood  5  ft.  10  in.  long  is  cut 
the  largest  possible  circle.     Find  its  area. 

16.  Find  the  area  of  a  sector,  if  the  radius  of  the  circle 
=  14  in.  and  the  angle  at  the  centre  =  45°.     The  sector  is 
the  same  part  of  the  circle  that  its  angle  is  of  360°. 

17.  Find  the  area  of  a  sector,  if  the  radius  =  10  in.  and 
the  angle  at  the  centre  =  18°. 


126 


LESSONS  IN  GEOMETRY. 


Lesson  66. 

1.  Theorem.—  In  every  right  triangle  the  square  of 
the  hypotenuse  is  equal  to  the  sum  of  the  squares 
of  the  legs. 

HYPOTHESIS. 

Let  ABC  be  a  right  triangle, 
and  let  Z  ACB  =  90°. 

CONCLUSION. 


E 


FIG. 133. 


PROOF.   Construct  the  squares 
ABDE,  ACFG,  BCHK. 

Draw  CM  _L  to  AB,  and  pro- 
duce it  to  meet  DE  at  N. 

Join  BG  and  CE. 

In  the  A  ACE,  ABG,  AC  =  AG,  AE  =  AB.    Why  ? 

Also,  Z  CAE  =  Z  BAG.    Why  ? 

Therefore,  A  ACE  =  A  ABG  (p.  66,  No.  14). 

Now,  the  A  ACE  and  the  rectangle  AMNE  have  the  same 
base,  AE,  and  the  altitude  of  each  is  the  distance  between 
the  parallels  AE  and  CN. 

Therefore,  A  ACE  =  %  rectangle  AMNE  (p.  120,  No.  1)  . 

Similarly,    A  ABG  =  %  square  ACFG. 

Therefore,  £  rectangle  AMNE  =  £  square  ACFG. 

Therefore,      rectangle  AMNE  =  square  ACFG. 

Similarly  (joining  CD  and  AK)  ,  we  can  prove  that 

rectangle  MBDN=  square  BCHK. 
Adding  the  last  two  equations,  we  have  : 

square  ABDE  =  square  ACFG  +  square  BCHK, 
or 


NOTE.  —  This  very  useful  theorem  is  called,  from  the  name  of  the 
discoverer,  the  "  Theorem  of  Pythagoras." 


AREAS.  127 

Find  the  hypotenuse  of  a  right  triangle,  having  given : 

3.  The  two  legs  3  in.  and  4  in. 

4.  The  two  legs  8  in.  and  15  in. 

5.  The  two  legs  81  ft.  and  108  ft. 

6.  The  two  legs  237  ft.  and  310  ft. 

Find  one  leg  of  a  right  triangle,  having  given  : 

7.  Hypotenuse  13  in.,  one  leg  5  in. 

8.  Hypotenuse  40  ft.,  one  leg  24  ft. 

9.  Hypotenuse  51  ft.,  one  leg  24  ft. 

10.  Hypotenuse  185  ft.,  one  leg  111  ft. 

11.  Find  the  area  of  a  right  triangle,  if  the  hypotenuse 
is  25  in.  and  one  leg  is  15  in. 

12.  Find  (correct  to  two  decimal  places)  the  hypotenuse 
of  a  right  triangle,  if  each  leg  is  10  in. 

13.  Find   (correct  to  two  decimal  places)  the  legs  of  an 
isosceles  right  triangle,  if  the  hypotenuse  is  10  in. 

14.  The  side  of  a  square  is  20  ft.     Find  the  diagonal. 

15.  The  diagonal  of  a  square  is  20  ft.     Find  one  side. 

16.  A   ladder  34  ft.    long  just  reaches  a  window  when 
placed  with  its  foot  16  ft.  from  the  side  of  the  house.     How 
high  is  the  window  above  the  ground? 

17.  Two  vessels  start  at  the  same  time  from  the  same 
port,  and  sail  one  north,  the  other  east.     Their  rates  are  G 
and  8  miles  per  hour.     How  far  apart  are  they  after  3  hours  ? 

18.  The  perimeter  of  an  isosceles  triangle  is  64  ft.  ;   the 
base  is  24  ft.     Find  the  area  of  the  triangle. 

19.  The  side  of  an  equilateral  triangle  is  14  yds.     Find 
the  area  of  the  triangle. 

20.  The  radius  of  a  circle  is  1  ft.  3  in.     Find  the  distance 
from  the  centre  to  a  chord  2  feet  long. 


128  LESSONS  IN   GEOMETRY. 


Lesson  67. 

1.  Define  the  mean  proportional  between  two  lengths. 
The  mean  proportional  between  two  lengths  is -the  length 

whose  square  is  equal  to  their  product. 

Thus :  since  6x6  =  4x9,  a  line  6  in.  long  is  the  mean 
proportional  between  a  line  4  in.  long  and  a  line  9  in.  long. 

2.  Problem.  —  To  find  the  mean   proportional    be- 
tween two  given  lines  AB  and  CD  (Fig.  1$4)> 

(1)  By  calculation.     Measure  the  two  lines,  multiply  their 
lengths  together,  and  find  the  square  root  of  the  product. 

(2)  By  construction.     Draw  a  straight  line,  and  upon  it 
lay  off  AB  —  a,  and  BC=b. 

Upon  AC  as  a  diameter,  describe 
a  semicircle. 

Erect  at  B  a  perpendicular,  cutting 
the  semicircle  at  D. 

Then  BD  will  be  the  mean  pro- 
portional required. 

NOTE.  —  This  construction  can  be  proved  to  be  correct  with  the  aid 
of  the  Theorem  of  Pythagoras,  but  the  shortest  and  easiest  proof  is 
founded  on  the  properties  of  similar  triangles  (see  page  151,  Ex.  8). 

3.  Draw  a  line  2  in.  long  and  a  line  3J  in.  long,  and  then 
find  (1)  by  calculation,   (2)  by  construction,  the  mean  pro- 
portional between  them. 

4.  Find   the   mean  proportionals  between  the  following 
numbers :  1  and  9,  9  and  25,  16  and  25,  1  and  ^,  %  and  ¥%. 

5.  What  is  meant  by  transforming  a  figure  ? 

To  transform  a  figure  is  to  change  its  shape  without  chang- 
ing its  size. 

6.  Construct  a  parallelogram,  and  then  transform  it  by  a 
construction  into  a  rectangle  (see  p.  118,  Ex.  1). 


AREAS. 


129 


FIG.  135. 


7.  Problem.  —  To    transform    a    rectangle    into    a 
square. 

(1)  By  calculation.     Measure  the  dimensions  of  the  rec- 
tangle ;    find  the  square  root  of  their  product ;  draw  a  line 
equal  to  this  root  in  length  ;    and  upon  this  line  as  a  side, 
construct  a  square. 

(2)  By  construction.   Denote  the 
two  dimensions  by  a  and  b. 

We  must  find  a  length  x  such 
that  x2  =  a  x  b. 

Hence  the  side  x  of  the  required 
square  is  the  mean  proportional 
between  a  and  b. 

Therefore  the  construction  is  the  same  as  that  in  No.  2. 

8.  Construct  a  rectangle  4  in.  long  and  2\  in.  wide.    Then 
find  (1)  by  calculation,  and  (2)  by  construction,  the  side  of 
the  equivalent  square. 

9.  How  can  a  triangle  be  transformed  into  a  square  ? 
Answer.     The  area  of  a  triangle  =  |-  the  base  X  altitude. 
Therefore,  the  side  of  an  equivalent  square  is  the  mean 

proportional  between  half  of  the  base  and  the  altitude. 

10.  Construct  a  triangle  with  base  4  in.  and  altitude  3^  in., 
and  transform  it  to  a  square  (1)  by  calculation  ;   (2)  by  con- 
struction. 

11.  Construct  an  equilateral  triangle,  and  then  transform 
it  by  construction  into  a  square. 

12.  Construct  a  rhombus,  and  then  transform  it  by  con- 
struction into  a  square  (see  p.  121,  No.  9). 

13.  Two  fields  are  equal  in  area.     One  is  a  square,  and 
the  other  a  rectangle  625  ft.  long  and  324  ft.  wide.     Find 
the  cost  of  fencing  each  field  at  10  cents  a  foot. 

14.  One  leg  of  an  isosceles  right  triangle  is  40  ft.  long. 
Find  (correct  to  two  decimals)  a  side  of  the  equivalent  square. 


130  LESSONS   IN   GEOMETRY. 


Lesson  68. 

1,  Problem.  —  To  transform  any  polygon  into  a  tri- 
angle. 

Let  ABODE  (Fig.  136)  be  any  pentagon. 

Draw  the  diagonal  BD. 

Draw  through  C  a  line  parallel  to  BD,  and  meeting  AB 
produced  at  F. 

Join  DF.  The  figure  AFDE  has  one  side  less  than  the 
given  polygon,  and  is  equivalent  to  it. 

Repeat  this  construction  by  joining  AD,  drawing  through 
E  a  line  parallel  to  AD,  and  joining  DG. 

The  A  DFG  will  be  equivalent  to  the  given  pentagon 
ABODE. 

The  repetition  of  this  process  will  reduce  a  polygon  of  any 
number  of  sides  to  an  equivalent  triangle. 


E 


G  A.  B  F 

FIG.  136. 

PROOF.  The  A  BDF,  BDC,  have  the  same  base  BD  and 
equal  altitudes  (their  vertices  C,  F,  being  in  a  line  II  to  BD) . 

Therefore  A  BDF  is  equivalent  to  A  BDC  (p.  120,  No.  2) . 

Hence,  if  A  BDC  be  taken  from  the  given  polygon,  and 
A  BDF  substituted  for  it,  the  polygon  will  suffer  no  change 
in  area,  but  it  will  be  transformed  into  the  figure  AFDE. 

For  like  reasons,  A  ADG  is  equivalent  to  A  ADE,  and 
when  substituted  for  it,  the  figure  AFDE  will  be  reduced  to 
the  A  DFG. 


AREAS.  131 

NOTE.  —  The  next  seven  exercises  are  to  be  done  by  construction. 

2.  Draw  a  trapezium,  and  transform  it  into  a  triangle. 

3.  Draw  a  pentagon,  and  transform  it  into  a  triangle. 

4.  Draw  a  hexagon,  and  transform  it  into  a  triangle. 

5.  Draw  a  hexagon,  and  transform  it  into  a  square. 

6.  Draw  a  triangle,  and  then  upon  one  side  of  this  tri- 
angle as  a  base,  construct  an  equivalent  isosceles  triangle. 

7.  Draw  a  square,  and  upon  a  side  of  the  square  as  a 
base,  construct  an  equivalent  isosceles  triangle. 

8.  Draw  a  triangle,  and  upon  one  of  its  sides  construct 
an  equivalent  rectangle. 

NOTE.  —  The  remaining  exercises  are  to  be  done  by  calculation. 

9.  The  legs  of  a  right  triangle  are  4  in.  and  8  in.     Find 
the  side  of  the  equivalent  square. 

10.  Find  the  side  of  a  square   equivalent   to  a  triangle 
whose  base  is  18  ft.  and  altitude  16  ft. 

11.  Find  the  side  of  a  square  equivalent  to  a  trapezoid 
whose  bases  are  30  ft.  and  34  ft.,  and  altitude  8  ft. 

12.  Find,   correct  to  two  decimals,  the  side  of  a  square 
equivalent  to  a  circle  whose  radius  is  7  in. 

13.  Find,  correct  to  two  decimals,  the  radius  of  a  circle 
equivalent  to  a  square  whose  side  is  11  in. 

14.  A  rectangle  is  320  ft.  long  and  100  ft.  wide.     If  the 
length  be  reduced  by  50  ft.,  how  much  must  the  breadth  be 
increased  in  order  that  the  area  may  remain  the  same  as 
before  ? 

15.  The  hypotenuse  of  an  isosceles  right  triangle  is  10  in. 
Find  (correct  to  two  decimals)  a  side  of  the  equivalent  square. 


132  LESSORS   IN   GEOMETRY. 

Lesson  69.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  60-65. 

2.  What  will  it  cost,  at  $2  per  rod,  to  fence  a  square 
field  containing  2^  acres? 

3.  How  many  five-acre  lots  can  be  made  out  of  a  field 
containing  5  square  miles? 

4.  How  many  planks,  each  10  ft.  long  and  8  in.  wide, 
will  cover  a  floor  14  ft.  3  in.  long  and  13  ft.  4  in.  wide? 

5.  Find  the  side  of  a  square  having  an  area  equal  to  that 
of  a  regular  hexagon  whose  side  is  8  ft. 

6.  The  circumferences  of  two  concentric  circles  are  440  ft. 
and  330  ft.,  respectively.     Find  the  width  of  the  ring. 

7.  Out  of  a  circular  piece  of  wood  whose  radius  is  3  ft. 
4  in.  is  cut  the  largest  possible  square.     Find  its  side. 

8.  The  radius  of  a  circle  is  4  ft.     Find  the  area  of  the 
inscribed  regular  hexagon  (see  p.  107,  No.  10). 

9.  The  side  of  a  square  is  42  yds.     Find  the  areas  of  the 
inscribed  and  the  circumscribed  circles. 

10.  The  diagonals  of  a  diamond-shaped  pane  of  glass  are 
12  in.  and  16  in.     How  many  panes  will  cover  an  area  of 
400  sq.  ft.  ? 

11.  A  square  and  a  rectangle  have  the  same  perimeter, 
100  yds.      The  length  of  the  rectangle  is    four  times   its 
breadth.    Which  has  the  greater  area,  and  by  how  much? 

12.  What  will  it  cost  to  turf  a  lawn  35  yds.  by  27  yds., 
with  turfs  21  in.  by  18  in.,  and  costing  3  cents  apiece? 

13.  A  path  8  ft.  wide  surrounds  a  rectangular  court  60  ft. 
long  and  36  ft.  wide.     If  tiles  are  9  in.  by  4  in.,  and  cost 
10  cents  each,  find  the  cost  of  paving  the  path  with  tiles. 


AREAS.  133 

Lesson  7O.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  66-68. 

2.  Show  by  a  figure  the  difference  between  half  a  square 
foot  and  half  a  foot  square. 

3.  Can  you  make  a  rectangle,  9  in.  by  4  in.,  and  then 
divide   it  into  two  parts  of  such  a  form  that  when  placed 
together  in  a  certain  way  they  will  make  a  square? 

4.  A  man  has  a  lot  of  land  127  yds.  square.     What  will 
it  cost  him,  at  30  cents  per  square  yard,  to  make  a  gravelled 
walk  1  yd.  wide  around  his  lot? 

5.  How  high  is  a  window  from  the  ground  if  a  ladder 
30  ft.  long  will  just  reach  the  window  when  placed  with  its 
foot  18  ft.  from  the  side  of  the  house? 

6.  The  width  of  a  house  is  47  ft.  ;  height  to  ridge,  67£  ft. ; 
height  to  eaves,  39  ft.     Find  the  cost  of  painting  the  side  of 
the  house  up  to  the  ridge,  at  18  cents  per  square  yard,  de- 
ducting 240  sq.  ft.  for  the  windows. 

7.  If  a  right  triangle  has  an  acute  angle  of  30°,  the  leg 
opposite  this  angle  is  always  equal  to  half  the  hypotenuse. 
Knowing  this  truth,  find  the  legs  of  a  right  triangle  if  the 
hypotenuse  =  20  ft.  and  one  angle  =  30°. 

8.  If  the  distance  from  the  centre  of  a  circle  to  a  chord 
10  in.  long  is  1  ft.,  find  the  distance  to  a  chord  2  ft.  long. 

9.  Transform  a  square  into  a  right  triangle. 

10.  Transform  a  rectangle  into  an  isosceles  triangle. 

11.  Transform  a  parallelogram  into  a  square. 

12.  Construct  a  square  equal  to  the  sum  of  two  given 
squares. 

13.  Construct  a  square  equal  to  the  difference  of  two  given 
squares. 


134  LESSONS   IN   GEOMETRY. 

Lesson  71.     Review. 

1.  Show  by  a  figure  that  the  square  of  £  is  equal  to  J. 

2.  Show  by  a  figure  that  the  square  of  1J  is" equal  to  2^. 

3.  Find  the  side  of  a  square  equal  in  area  to  the  sum  of 
the  squares  constructed  upon  the  lengths  2  in.,  3  in.,  and  4  in. 

4.  In  a  railway  curve  the  radius  of  the  circle  is  300  ft., 
and  the  centre  of  the  circle  is  located  at  a  point  780  ft.  from 
a  station.     Draw  a  plan  showing  the  location  of  the  station, 
the  centre  of  the  circle,  and  the  point  where  the  track  begins 
to  curve.     Find  the  distance  from -this  point  to  the  station. 

5.  How  many  bricks  9  in.   by  4J  in.   are   required   to 
cover  a  square  yard? 

6.  Transform  a  triangle  into  a  rectangle. 

7.  A  circular  grass-plot  has  a  diameter  of  400  ft.,  and  a 
walk  3  yds.  wide  around  it.     Find  the  area  of  the  grass-plot, 
and  also  the  area  of  the  walk. 

8.  A  square,  whose  side  is  4  in.,  is  to  be  transformed 
into  a  rhombus  in  which  one  diagonal  equals  twice  the  other. 
Find  the  lengths  of  the  diagonals  and  of  one  side  of  the 
rhombus.     Can  you  solve  this  problem  also  by  construction? 

9.  Find  the  radius  of  a  circle  equivalent  to  a  square 
whose  side  is  11  ft.  long. 

10.  Find  the  side  of  a  square  equivalent  to  a  circle  whose 
radius  is  7  ft.  long. 

11.  Find  the  area  of  a  trapezium,  if  one  diagonal  is  54  ft., 
and  the  perpendiculars  to  this  diagonal  from  the  opposite 
vertices  are  23  ft.  9  in.  and  36  ft.  6  in. 

12.  Compare  the  areas  of  a  circle,  a  square,  and  an  equi- 
lateral triangle,  if  the  perimeter  of  each  figure  is  132  ft. 


AREAS.  135 

Lesson  72.    Review. 

1 .  Review  all  the  italicized  exercises  in  this  chapter. 

2.  Find  the  area  of  a  right  triangle,  if  the  hypotenuse 
is  8J  in.  and  one  of  the  legs  is  4  in. 

3.  Find  the  area  of  an  isosceles  triangle,   if  the  base 
is  20  ft.  and  each  of  the  other  sides  is  26  ft. 

4.  Draw  any  pentagon,  and  transform  it  into  a  square. 

5.  Construct  a  square  whose  diagonal  shall  be  3  in.,  and 
then  find  its  area. 

6.  Make  a  square  half  as  large  as  a  given  square. 

7.  Find  the  side  of  a  square  equivalent  to  a  rectangle 
400  ft.  long  and  100  ft.  wide. 

8.  One  angle  of  a  rhombus  is  60°,  and  the  shorter  diag- 
onal is  8  ft.     Find  the  area  (see  p.  133,  No.  7). 

9.  The  angle  of  a  sector  is  60°,  and  the  chord  of  the 
arc  is  14  ft.     Find  the  area  of  the  sector. 

10.  Two  ships,  each  making  6  miles  an  hour,  sail  from 
the  same  port.     One  sails  at  noon,  directly  east ;  the  other 
sails  at  3  P.M.,  directly  south.      How  far  apart  will  they 
be  at  6  P.M. ? 

11.  The  top  of  a  centre-table  has  the  shape%of  a  regular 
octagon.     One  side  measures  2  ft.    What  will  it  cost  to  stain 
and  polish  the  surface  at  $2.25  per  square  foot? 

12.  A  room  is  31£  ft.  by  17  ft.,  and  12£  ft.  high,  with  3 
windows  each  5^  ft.  by  6  ft.,  2  doors  each  7£  ft.  by  4  ft., 
and  a  fireplace  7  ft.  by  6£  ft.     The  carpet  is  27  in.  wide, 
costs  $2.50  per  yard,  and  is  laid  lengthwise  along  the  floor. 
The  paper  is  27  in.  wide,  and  costs  $1.80  per  roll  of  10  yds. 
Find  the  cost  of  the  carpet  and  the  paper. 


136  LESSONS   IN   GEOMETRY. 

Lesson  73.    Review. 

1.  Define  and  name  the  metric  units  of  area. 

The  metric  units  of  area  are  the  squares  of- the  units  of 
length.  They  are  named  by  placing  the  word  "square" 
before  the  name  of  the  corresponding  unit  of  length.  Thus  : 
square  meter,  square  decimeter,  etc. 

The  square  dekameter  is  also  called  an  ar,  and  the  square 
hectometer  a  hectar. 

The  abbreviations  are  formed  by  placing  the  letter  q  be- 
fore those  of  the  corresponding  units  of  length.  Thus  :  qm 
for  square  meter ;  qcm  for  square  centimeter,  etc. 

2.  How  do  they  stand  related  to  each  other? 

Each  unit  is  100  times  larger  than  the  next  smaller  one. 
For  example  :  lqm  =  100qdm  ;  lqdm  =  100qcm,  etc. 

3.  Show  by  a  figure  that  1  square  meter  =  100qdra. 

4.  Reduce  to  square  meters  4dkm  ;  l,840,000qmm. 

5.  How  many  square  meters  are  there  in  a  hectar? 

6.  Show  that  lqm  =  10.764  sq.  ft.  (see  bottom  of  p.  33). 

7.  Show  that  there  are  about  15^  sq.  in.  in  lqdm. 

8.  A  square  park  contains  64  hectars.     How  many  trees, 
20m  apart,  can  be  set  out  around  it  ? 

9.  If  glass  is  worth  $1.40  per  square  meter,  what  will  be 
the  cost  of  1000  panes  of  glass  each  48cm  square? 

10.  Find  the  area  of  the  largest  rectangle  which  you  can 
make  under  the  condition  that  its  perimeter  is  60cm. 

11.  A  table  2.3m  by  1.2m  is  covered  with  cloth  which  over- 
laps 3cra  on  all  sides.     How  much  cloth  was  required? 

12.  Find  the  side  of  a  square  equivalent  to  a  triangle 
whose  base    is  90ra  and  altitude  is  20m. 


AREAS. 


137 


13.  A  ladder  13m  long  leans  against  the  top  of  a  wall,  its 
foot  being  5m  from  the  bottom.     How  high  is  the  wall? 

14.  Find  the  area  of  an  equilateral  triangle  if  the  length 
of  one  side  is  20m. 

15.  Find  the  cost  of  covering  a  round  table  with  cloth 
60cm  wide,  and  costing  $2  per  meter,  if  the   radius  of  the 
table  is  equal  to  75cm. 

16.  Find  the  cost  of  making  a  stone  walk  lm  wide  around 
a  circular  reservoir  64m  in  diameter,  the  price  paid  being  $5 
per  square  meter. 

17.  At  a  point  E  in  one  side  of  the  rectangular   field 
ABCD  there  is  an  artesian  well.     The 

owner  divides  the  field  into  three  unequal 
parts  by  lines  drawn  from  E  to  D  and 
C.  Find  the  value  of  each  part,  if 
AD  =  AE  =  81m,.  EB  =  144m,  and  a 
hectar  is  worth  $3400. 


FIG.  137. 


18.  The  radius  of  a  circle  is  4in.     Find  the  areas  of  the 
circumscribed  and  the  inscribed  squares. 

19.  In  order  to  find  the  area  of  a  swamp,  ABCDEF,  it 
was  first  surrounded  by  a  rectangle,  as  shown  in  Fig.  138. 

The    following   measurements 
were  then  made  : 


G 


B 


CH=75.om, 

DI  =68.5m,  IE  =  68.5 
KF=  29.  5m,  FA  =  58.8n 

Find  the  value  of  the  swamp,     K 
one  hectar  being  worth  $1500. 

20.    The  diagonals  of  a  rhombus  are  4m  and  6m.     Find  its 
area,  its  perimeter,  and  its  altitude. 

OF  THE^  " 

{    UNIVERSITY 


138  LESSONS    IN    GEOMETRY. 

Lesson  74.    Review. 

1.  Review  all  the  italicized  exercises  in  this  chapter. 

2.  Find  the  area  of  the  figure  which  is  given  to  you. 

3.  If  a  triangle  and  a  parallelogram  have  equal  bases  and 
equal  areas,  what  is  true  of  their  altitudes? 

4.  Find  the  radius  of  a  circle  three  times  as  large  as  a 
circle  whose  radius  is  7m. 

5.  Show  by  a  figure  that  the  square  of  |  is  equal  to  1. 

6.  The  three  dimensions  of  a  room  are  each  6. 4m.     Find 
the  cost  of  papering  the  walls  at  30  cents  per  square  meter. 

7.  Through  the  middle  of  a  rectangular  garden,  180m  by 
76m,  there  are  two  paths  _L  to  each  other  and  II  to  the  sides. 
Find  the  area  of  the  remainder  of  the  garden. 

8.  If  a  rhombus  and  a  square  have  equal  perimeters, 
which  has  the  greater  area?     Prove  your  answer  correct. 

9.  Find  the  side  of  a  square  equivalent  to  a  trapezoid 
whose  bases  are  60m  and  32m,  and  altitude  is  124m. 

10.  Find  the  side  of  a  square  equivalent  to  a  regular  hexa- 
gon, one  side  of  which  is  8m. 

11.  The  perimeter  of  a  regular  octagon  is   16m.      Find 
the  area. 

12.  One  leg  of  an  isosceles  right  triangle  is  32m.     Find 
the  area. 

13.  The  diagonal  of  a  square  is  12m.     Find  the  area. 

14.  Find  the  length  of  the  shortest   chord  that  can  be 
drawn  through  a  point  9cm  from  the  centre  of  a  circle  whose 
radius  is  15cm. 

15.  What  is  the  mean  proportional  between  9  and  16? 

16.  The  angle  of  a  sector  is  120°,  and  the  length  of  its 
arc  is  66cm.     Find  its  area. 


CHAPTER  VIII. 


RATIOS. 

Lesson  75. 

1.  Define  the  numerical  measure  of  a  quantity. 

The  number  of  times  one  quantity  contains  another  quan- 
tity of  the  same  kind  is  called  the  numerical  measure  of 
the  first  quantity  referred  to  the  second  as  a  unit. 

ABCDEFGH 

i i i i i i i i 

For  example  :  if  AB  =  BC  =  CD  =  DE  =  EF = FG  =  GH, 
the  numerical  measure  of  AH  referred  to  AB  as  a  unit  is  7, 
and  the  numerical  measure  of  AB  referred  to  AH  as  a  unit 
is  | ;  in  other  words,  AH=  1  AB,  AB  =  i  of  AH. 

2.  If  in  the  above  figure  AB  is  taken  as  the  unit,  what  is 
the  numerical  measure  of  AG?  AF?  AE?  AD?  AC?  AB? 

3.  If  AC  is  taken  as  the  unit,   what  is  the   numerical 
measure  of  AH?  AG?  AF?  AE?  AD?  AC?  AB? 

4.  Define  the  ratio  of  two  numbers. 

The  ratio  of  one  number  to  another  number  is  their  rela- 
tive magnitude,  and  is  found  by  dividing  the  first  number  by 
the  second  number. 

A  ratio  is  written  either  as  a  fraction  or  by  placing  a 
colon  (  :  )  between  the  two  numbers.  For  example  : 

the  ratio  of  4  to  7  is  ^,  or  4  :  7  ; 

the  ratio  of  3  to  1  is  f  (that  is,  3),  or  3  : 1. 

The  two  numbers  which  form  a  ratio  are  called  its  terms. 


140  LESSONS   IN   GEOMETRY. 

5.  Define  the  ratio  of  two  quantities. 

The  ratio  of  two  quantities  of  the  same  kind  is  the  ratio 
of  their  numerical  measures  referred  to  the  same  unit. 

(1)  The  two  quantities  must  be  the  same  in  kind.     Two 
quantities  which  differ  in  kind  cannot  have  a  ratio  to  each 
other.     For  example,  we  cannot  compare,  as  regards  magni- 
tude, a  line  with  an  angle,  or  dollars  with  days. 

(2)  The  two  quantities  must  be  referred  to  the  same  unit. 
We  may  compare,  for  example,  6  in.  with  2  ft.,  but  their 
ratio  is  not  6  :  2,  because  1  ft.  =  12  in.  ;  it  is  6  :  24,  or  1  :  4. 

ABCDEFGH 

6.  In  the  above  figure  what  is  the  ratio  of  AC  to  AD? 
AD  to  AC?  AC  to  AH?  AD  to  AG?  AE  to  AG?  etc. 

7.  What  is  the  ratio  of  2  in.  to  6  in.  ?  2  in.  to  6  ft.  ? 

8.  What  is  the  ratio  of  15°  to  a  right  angle? 

9.  Problem.  —  To  divide  a  straight  line  AB  into  two 
parts  having  a  given  ratio  ;  say,  3  :  4- 

Draw   through  A  a  line  AC, 
making  an  acute  angle  with  AB. 

Begin  at  A  and  lay  off  upon 
AC  3  +  4,  or  7,  equal  parts. 

Mark  the  end  of  the  7th  part  D, 
and  the  end  of  the  3d  part  E. 

Join  BD,  and   draw  EF  II  to   ^~         ~~p~  B 

BD,  and  cutting  AB  at  F.  FlG  139 


PROOF.  Draw  parallels  to  BD  through  all  the  points  of 
division  in  AC,  and  prove  that  they  divide  AB  into  3+4, 
or  7,  equal  parts  (p.  25,  No.  4;  p.  86,  No.  4). 

Therefore  AF  :  FB  =  3  :  4  . 

10.    Divide  a  line  into  two  parts,  having  the  ratio  2  :  5. 


RATIOS. 


141 


11.  Compare  two  triangles  whose  altitudes  are  equal. 
Let  the  triangles  ABC,  ADC  (Fig.  140),  have  the  same 
altitude  CE,  but  'let  their  bases  AB,  AD,  be  unequal. 
The  areas  of  the  two  triangles  have  the  values, 


If  we  divide  one  of  these  equations  by  the  other,  we  obtain 
the  ratio  of  the  areas  written  as  a  fraction  ;  this  fraction  may 
then  be  reduced  to  its  lowest  terms  by  cancelling  the  common 
factors  ^  and  CE.  We  have  as  the  result, 

A  ABC  =  $ABx  CE^AB 
A  ADC     ^ADxCE     AD 

Hence  we  see  that  triaixjles  having  equal  altitudes  are  to 
each  other  as  their  bases. 

c 


FIG. 141. 

12.  Show  that  two  triangles  having  equal  bases  are 
to  each  other  as  their  altitudes  (Fig.  14D- 

13.  Compare  two  parallelograms  having  equal  alti- 
tudes. 

14.  Compare  two  parallelograms  having  equal  bases. 

15.  Two  parallelograms  have  equal  altitudes,  but  the  base 
of  one  is  ten  times  that  of  the  other.     Compare  their  areas. 

16.  A  triangle  and  a  parallelogram  have  equal  bases,  but 
the  altitude  of  the  triangle  is  four  times  that  of  the  parallelo- 
gram.    What  is  the  ratio  of  their  areas? 

17.  Construct  a  square  and  a  rhombus  so  that  their  perim- 
eters shall  be  equal,  and  their  areas  be  to  each  other  as  3  :  2. 


142 


LESSONS  IN  GEOMETRY. 


Lesson  76. 

1.  Compare  the  two  parts  into  which  a  triangle  is 
divided  Inj  drawing  a  line  from  one  vertex,  to  the 
opposite  side. 

In  the  triangle  ABC  (Fig.  142)  let  CD  be  drawn  from  the 
vertex  C  to  any  point  D  of  the  base  AB.  The  two  triangles 
thus  formed  ADC,  BDC,  have  the  same  altitude  CE,  but 
unequal  bases  AD,  BD.  Therefore  (p.  141,  No.  11), 

A  ADC =  AD 
A  BDC  ~  BD 


A  ADC:  A  BDC  =  AD  :  BD,  or 


Hence  a  line  draivn  from  the  vertex  to  the  base  of  a  tri- 
angle divides  the  triangle  and  its  base  into  parts  that  have  the 
same  ratio. 

C 


D        E 

FIG.  142. 


B 


E 


FIG.  143. 


2.  How  must  the  line  CD  (Fig.  142)  be  drawn  in  order  to 
divide  A  ABC  into  two  equivalent  parts?     What  would  be 
the  ratio  of  these  parts  ?     What  would  be  the  ratio  of  one  of 
them  to  the  whole  triangle  ? 

3.  Draw  a  triangle  and  divide  it  into  two  parts  which  shall 
be  to  each  other  as  2:3.     What  part  of  the  whole  triangle  is 
each  one  of  the  parts  ? 

4.  It  is  desired  to  run  a  division  line  AE  across  a  rec- 
tangular field  ABCD  (Fig.  143)  so  that  the  triangle  cut  off, 
ADE,  may  be  equal  in  area  to  ^  of  the  whole  field.     How 
would  you  determine  the  point  E?    What  part  of  DC  is  DE? 
What  is  the  ratio  of  DE  to  EC? 


KATIOS.  143 

5.  In  Fig.  143,  if  DE  =  $DC,  what  is  the  ratio  of  DE 
to  DC?  DC  to  DE ?  DE  to  EC?    What  part  of  the  rectangle 
ABCD  is  the  A  ADE? 

How  should  AE  (Fig.  143)  be  drawn  in  order  that 

6.  A  ADE  may  be  \  of  the  rectangle  ABCD? 

7.  A  ADE  may  be  1  of  the  rectangle  ABCD? 

8.  A  ADE  :  rectangle  ABCD  =  1:12? 

9.  A  ADE  :  rectangle  ABCD  =  3:7? 

10.  Draw  a  triangle  and  divide  it  into  three  parts  which 
shall  be  to  each  other  as  the  numbers  1,  2,  3. 

11.  Draw  a  rectangle  and  then  divide  it  into  four  equiva- 
lent parts  by  lines  drawn  from  one  of  the  vertices. 

12.  Draw  a  rectangle  and  then  divide  it  into  three  equiva- 
lent parts  by  lines  drawn  from  one  of  the  vertices. 

13.  Draw  a  rectangle  and  then  divide  it  by  a  line  II  to  one 
side  into  two  parts  having  the  ratio  2:3.     What  fraction  of 
the  rectangle  is  each  part? 

14.  In  a  triangle  ABC  (Fig.  144)  the  middle  points  Z>,  E, 
of  the  sides  AB,  AC,  are  joined.     What  is  the  ratio  of  the 
triangle  ADE  to  the  triangle  ABC? 

Join  BE.     Since  A  D  =  M  B,  A 

A  ADE  =£A  ABE  (p.  141,  No.  11). 
Also,  since  AE=  I  AC, 


Substituting  this  value  of  A  ABE  in  the 
first  equation,  we  have  : 


FIG.  144. 
or,  written  as  ratios,  A  ADE  :  A  ABC=l  :  4. 


15.    In  Fig.  144,  if  AD  =  $AB,  and  AE  =  $AC,  show 
that  A  ADE  :  A  ABC  =  1:9. 


144  LESSONS  IN   GEOMETKY. 

Lesson  77. 

1.  Define  a  proportion. 

Four  quantities  are  said  to  be  proportional,  or  to  form  a 
proportion,  if  the  ratio  of  the  first  to  the  second  is  equal  to 
the  ratio  of  the  third  to  the  fourth. 

For  example:  since  T\  =  f,  the  numbers  4,  10,  2,  5,  are 
proportional. 

A  B  E  F 


D       G  If 


Again,  the  lines  AB,  CD,  have  the  ratio  3  :  5,  and  the 
lines  EF,  GH,  have  the  same  ratio  3:5.  Therefore  the  four 
lengths  AB,  CD,  EF,  GH,  form  the  proportion 

AB  :  CD  =  EF  :  GH, 

which  is  read  "AB  is  to  CD  as  EF  is  to  GH." 

The  four  quantities  that  form  a  proportion  are  called  its 
terms  ;  the  first  and  fourth  are  called  the  extremes  ;  the  second 
and  third  the  means. 

2.  Give  an  example  of  four  proportional  numbers. 

3.  Give  examples  of  proportions  on  pp.  142,  143. 

4.  Prove  that,  if  four  numbers  form  a  proportion, 
the  product  of  the  extremes  is  equal  to  the  product 
of  the  means. 

If  the  four  numbers  a,  6,  c,  d,  are  proportional,  then 


Multiply  both  sides  of  this  equation  by  the  product  ltd. 
abd      cbd 

Then  T--5" 

Cancelling  common  factors,  we  have  left  ad  =  be. 


RATIOS.  145 

5.  Find  the  first  term  of  a  proportion  if  the  other 
three  terms  are  9,  28,  7. 

Let  x  denote  the  first  term  ;  then  x  :  9  =  28  :  7. 

Hence  (No.  5),    7z  =  9  x  28. 
q  v  90 
Therefore  x  =  ^4^  =  9  x  -  =  36. 

In  general,  either  extreme  is  equal  to  the  product  of  the 
means  divided  by  the  other  extreme. 

6.  Find  the  first  term  if  the  other  terms  are  3,  7,  84. 

7.  Find  the  fourth  proportional  to  13,  65,  120. 

8.  Compare  two  circumferences  and  their  radii. 
Let  c,  d,  denote  the  circumferences  ;  r,  s,  their  radii. 

Then  c  =  27rr,  d  =  '27rs  (p.  108,  No.  1). 

Therefore  <L  =  ^  =  L 

d      2  TTS      s 

Or,  two  circumferences  are  to  each  other  as  their  radii. 

9.  Compare  the  areas  of  two  circles    and    their 
radii. 

Let  a,  6,  denote  the  areas  of  two  circles  ;  r,  s,  their  radii. 
Then  a  =  irr8,  b  =  irs2  (p.  124,  No.  1) . 

Therefore  ft  =  ^  =  £ 

b          TTS2          S3 

In  other  words,  the  areas  of  two  circles  are  to  each  other  as 
the  squares  of  their  radii. 

10.  How  is  the  circumference,   and  also   the  area,  of  a 
circle  changed  if  the  radius  is  doubled?  trebled?  halved? 

11.  The  radii  of  two  circles  are  as  2:5.     The  area  of  the 
first  circle  is  100  sq.  ft.    What  is  the  area  of  the  other  circle? 

12.  The  radii  of  three  circular  tanks  are  6  ft.,  9  ft.,  and 
15  ft.     If  a  mason  charge  So  for  cementing  the  smallest  tank, 
what  ought  he  to  charge  for  cementing  each  of  the  others  ? 


146  LESSOHS  IN  GEOMETRY. 

Lesson  78. 

1.  Define  similar  figures  (see  p.  64,  No.  3A 
Similar  figures  are  figures  that  have  the  same-shape. 
For  example,  all  circles  are  similar  figures. 

2.  Define  similar  triangles. 

Two  triangles  have  the  same  shape  if  their  angles  are 
equal,  each  to  each ;  hence  we  may  define  similar  triangles 
as  triangles  that  are  equiangular  ivith  respect  to  each  other. 

3.  Explain  by  reference  to  Fig.  145  what  is  meant  by  two 
triangles  being  equiangular  with  respect  to  each  other. 

4.  Name  by  pairs  the  homologous  sides  of  the  triangles  in 
Fig.  145.     For  the  meaning  of  homologous,  see  p.  65,  No.  4. 


A  B       7)  E  A          I)          F          H  B 

FIG.  145.  FIG.  146. 

5.  What  is  the  most  important  property  which  two 
similar  triangles  have? 

The  homologous  sides  of  tivo  similar  triangles  are  propor- 
tional. 

NOTE.  —  This  property  can  be  deduced  as  a  necessary  consequence 
of  the  definition  in  No.  2,  but  the  proof  will  not  be  given  here. 

6.  One  proportion  formed  by  the  homologous  sides   in 
Fig.  145  is  AB:DE  =  BC  :  EF.     State  the  two  others. 

7.  If  AB  =  15  ft.,  DE  =  $  ft.,  50  =  20  ft.,  find  EF. 

8.  If  AC=  10  ft.,  BC  =  20  ft.,  DF=  G  ft.,  find  EF. 

9.  If  (Fig.  145)  AB  is  three  times  DE,  wnat  is  true  of 
AC  and  DF?  of  BC  and  EF? 


147 

10.  Between  the  sides  AB,  AC  of  the  triangle  ABC 
(Fig.  146)  a  line  DE  parallel  to  BC  is  drawn.    Com- 
pare the  two  triangles  ABC,  ADE. 

The  Z  A  is  common  to  the  two  triangles. 

Z  ADE  =  Z  ABC,  and  Z  AED  =  Z  ACB.    Why  ? 

Therefore  the  triangles  ABC,  ADE  are  similar  (No.  4) . 

11.  The  side  AB  of  A  ABC  (Fig.  146)  is  divided  into  four 
equal  parts,  and  through  the    points  of  division  D,  F,  H, 
parallels  to  BC  are  drawn,  meeting  AC  at  E,  G,  K,  respec- 
tively.     How  many  similar  A  are  formed?     Why  similar? 
State  several  proportions  formed  by  their  homologous  sides. 

12.  Let  (Fig.  146)  AD  =  U  ft.,  AE  =  18  ft.,  DE  =  9  ft. 
Find  AB,  AC,  BC. 

13.  Prove  that  AE  =  EG  =  GK=  KG.     (Two  methods.) 

14.  Draw  any  line  AB,  and  then  construct  a  length  CD 
so  that  CD  :  AB=  5  :  3  (see  No.  10,  and  p.  140,  No.  9). 

15.  Draw  any  triangle  ABC,  and  construct  a  similar  tri- 
angle so  that  the  ratio  of  any  two  homologous  sides  shall  be 
equal  to  that  of  the  numbers  4  and  7. 

16.  Define  ratio  of  similitude. 

The  ratio  of  similitude  of  a  triangle  with  respect  to  a  simi- 
lar triangle  is  the  ratio  of  any  one  of  its  sides  to  the  homol- 
ogous side  of  the  similar  triangle. 

-  If  (Fig.  145)  AB  =  lo  ft.,  and  DE  =  V  ft.,  then  the  ratio 
of  similitude  of  A  ABC  with  respect  to  A  DEF  is  -|,  or  5  :  3, 
and  that  of  A  DEF  with  respect  to  A  ABC  is  f ,  or  3  :  5. 

17.  What  is  the  ratio  of  similitude  of  A  ABC  (Fig.  146) 
with  respect  to  A  ADE  ?  A  ABC  with  respect  to  A  AFG  ?  etc. 

The  sides  of  a  triangle  are  12,  16,  20.  Find  the  sides  of 
a  similar  triangle  if  its  ratio  of  similitude  is 

18.  2:  1.  20.    1:4.  22.    2:3.  24.    1:20. 

19.  1:2.  21.    5:2.  23.    4:3.  25.    100:1. 


148 


LESSONS   IN   GEOMETRY. 


Lesson  79. 

1.  How  can  the  height  of  a  vertical  object  be  found 
by  means  of  its  shadow  ? 

Let  the  tree  AB  (Fig.  147)  be  the  object.  Fix  a  rod  DE 
vertically  in  the  ground  near  the  tree.  Measure  DE  and  the 
lengths  AC,  DF  of  the  shadows  cast  by  the  tree  and  the  rod. 
Since  rays  of  light  are  parallel  to  one  another,  the  triangles 
ABC,  DEF  are  equiangular  with  respect  to  each  other  ; 
hence  they  are  similar  ;  hence  the  shadows  cast  by  the  two 
objects  are  proportional  to  the  heights  of  the  objects.  There- 
fore if  x  denote  the  height  of  the  tree  AB, 


DExAC 


E 


A 


C  D       F 


FIG.  147. 


FIG.  148. 


2.  A  tree  casts  a  shadow  78  ft.  long  at  the  same  time  that 
a  rod  8  ft.  high  casts  a  shadow  G^  ft.  long.     How  high  is 
the  tree? 

3.  Under  what  condition  would  the  length  of  the  shadow 
AC  (Fig.  147)  be  just  equal  to  the  height  AB  of  the  tree? 

4.  Explain  how  you  would  proceed  in  order  to  find  the 
breadth  AB  of  a  river  (Fig.  148)  by  means  of  two  similar 
triangles  ABC,  EDC. 

If  AC=  300  ft.,  CD  =  50  ft.,  ED=  150  ft.,  find  AB. 

5.  If  J2Z)=500ft.,  zndAC=2CD,  find 


RATIOS. 


149 


6.  In  what  way  are  the  areas  of  two  similar  tri- 
angles related  to  their  homologous  sides? 

The  relation  is  expressed  by  the  following  theorem  : 
The^areas  of  two  similar  triangles  are  proportional  to  the 
squares  of  any  tivo  homologous  sides. 

The  proof  of  this  theorem  will  not  be  given  here. 

7,  Illustrate  the  meaning  of  this  theorem. 
Draw  a  A  ABC  (Fig.  149),  divide 

each  side  into  4  equal  parts,  and 
draw  through  the  points  of  division 
lines  II  to  the  sides.  Several  similar 
A  will  be  formed.  Why  similar? 
Also  the  A  ABC  will  be  divided  into 
16  equal  A  in  such  a  way  that  the 
ratio  of  the  areas  of  any  two  similar 
A  is  seen  almost  at  a  glance. 

Compare,  for  example,  the  A  ABC,  AHK.  The  ratio  of 
their  areas  is  seen  to  be  16:9,  or  equal  to  the  square  of  the 
ratio  4  :  3  of  any  two  homologous  sides. 

8.  Compare  the  areas  and  the  homologous  sides  of  the 
A  ADE,  AFG  ;  the  A  ADE,  AHK]  the  A  AFG,  AHK,  etc. 

9.  The  homologous  sides  of  two  similar  triangles  are  3  ft. 
and  7  ft.     What  is  the  ratio  of  their  areas  ?     If  the  area  of 
the  first  triangle  is  36  sq.  ft.,  find  the  area  of  the  second 
triangle. 

10.  Compare  the  areas  of  two  similar  triangles  having 
a  and  b  for  homologous  sides,  if  a  =  26;  a  =36;  a  = 


N       P 
pIG.  149, 


11.  The  side  of  a  A=  3  ft.     Find  the  homologous  side  of 
a  similar  A  25  times  as  large  ;  also  of  one  half  as  large. 

12.  It  is  desired  to  divide  a  A  ABC  into  two  equivalent 
parts  by  drawing  a  line  DE  II  to  BC.      If  AB  =  400  ft., 
AC=  500  ft.,  BC=  600  ft.,  find  AD  and  AE. 


150 


LESSONS   IN    GEOMETRY. 


Lesson  8O. 

1.  Show  that  two  triangles  are  similar  if  they  have 
two  angles  equal  each  to  each. 

Apply  No.  3,  p.  63,  and  No.  2,  p.  146. 

2,  Prove  that  in  a  right  triangle  the  perpendicular 
dropped  from  the  vertex  of  the  right  angle  to  the 
hypotenuse  divides  the  triangle  into  two  right  tri- 
angles which  are  similar  to  each  other  and  to  the 
entire  triangle  (Fig.  150). 

In  the  A  ABC  let  c 

BC  =«,  CD  be  _L  to  AB, 


=  c,  Z.BAC=r, 
=  d,  /.ABC=s, 


D     c 

FIG.  150. 


B 


CD  =  h. 

Show  that  Z.ACD  =  s,  and  Z  BCD  =  r,  and  then  apply 
No.  2,  p.  146,  or  No.  1,  above. 

3.  In  Fig.  150  a  right  triangle  is  divided  into  two  similar 
right  triangles,  as  explained  in  No.  2,  and  the  sides  and 
angles  are  named  by  small  letters,  as  shown  in  the  figure. 
Complete  the  following  table  by  writing  with  each  angle 
the  homologous  sides  of  the  three  triangles. 


A  ABC 

&HCI> 

&ACD 

90° 

r 

s 

RATIOS.  151 

4.  Six  different  proportions  may  be  formed  from  the 
homologous  sides  of  the  similar  triangles  in  Fig.  150,  and  it 
is  very  easy  to  form  them  after  the  table  in  No.  3  has  been 
completed.  Write  out  all  these  proportions  in  a  vertical 
column,  thus : 

c:  a  =  a  :  e, 

c  :  a  =  b  :  h. 


5.  Transform  the  six  proportions  obtained  in  No.  4  into 
c'(iuations  by  placing  in  each  case  the  product  of  the  means 
equal  to  the  product  of  the  extremes. 

6.  Define  a  mean  proportional  (p.  128,  No.  1). 
The  proportion  c  :  a  —  a  :  e  gives  the  equation  a2  =  ce. 
Between  what  two  lines  is  a  a  mean  proportional? 

7.  What  two  other  examples  of  mean  proportionals  are 
there  among  the  equations  obtained  in  No.  5,  ? 

8.  Prove  that  the  construction  in  No.  2,  p.  128,  is  correct. 

HINTS.  —  First  show  that  the  triangle  ACD  (Fig.  134)  is  a  right 
triangle ;  then  make  use  of  one  of  the  proportions  which  can  be  obtained 
from  the  table  in  No.  3. 

9.  Among  the  equations  obtained  in  No.  5  are  the  two 
following  :    a2  =  ce,  b2  =  cd.      Hence  show  that  a2  -f-  b2  =  c2. 
State  this  result  as  a  theorem.     Compare  with  No.  1,  p.  126. 

HINT.  —  Observe  that  d  -f  e  =  c. 

10.  If  (Fig.  150)  c  =  9  in.,  e  =  4  in.,  find  a,  h,  and  b. 

11.  If  (Fig.  150)  a  =  6  in.,  b  =  8  in.,  find  c,  d,  and  e. 

12.  If  (Fig.  150)  e  =  3  in.,  h  =  4  in.,  find  a,  6,  and  c. 

13.  One  of  the  equations  obtained  in  No.  5  is  the  follow- 
ing :  ab  =  ch.     State  this  equation  as  a  theorem. 


152  LESSONS    IN   GEOMETRY. 

Lesson  81.    Review. 

1.  Review  the  italicized  exercises  in  LCSSODS  75-77. 

2.  What  is  the  ratio  of  3  inches  to  2  feet  ? 

3.  What  is  the  ratio  of  6  feet  to  9  yards  ? 

4.  What  is  the  ratio  of  £  of  an  inch  to  ^  of  an  inch  ? 

5.  What  is  the  ratio  of  \  of  an  inch  to  J  of  a  foot? 

6.  Produce  a  straight  line  AB  to  a  point  C  such  that 
AB :  BO  =  4  :  5.    What  is  the  ratio  of  AC  to  BC? 

7.  Divide  a  given  line  AB  into  three  parts  proportional 
to  the  numbers  2,  3,  5. 

8.  Divide  a  given  line  AB  into  three  parts  proportional 
to  the  numbers  1,  ^,  ^. 

9.  Construct  a  triangle  whose  sides  shall  be  to  each  other 
as  the  numbers  3,  4,  5. 

10.  Construct  an  isosceles  triangle  so  that  the  ratio  of  the 
base  to  one  leg  shall  be  as  2:3. 

11.  Compare  the  areas  of  a  triangle  and  a  parallelogram 
if  they  have  equal  bases  and  equal  altitudes. 

12.  A  triangle  and  a  parallelogram  have  equal  areas  and 
equal  altitudes.     What  is  the  ratio  of  their  bases  ? 

13.  Divide  a  triangle  into  two  parts  which  are  as  3  to  7. 

14.  A   triangular  lot  is  worth  $6000.     Show  by  a  figure 
how  you  would  run  a  line  from  one  corner  to  the  opposite 
side  so  as  to  cut  off  a  piece  worth  $2400. 

15.  Divide  a  parallelogram  by  a  line   drawn   from   one 
vertex  into  two  parts  having  to  each  other  the  ratio  3:8. 

16.  Find  the  fourth  proportional  to  the  numbers  9,  117, 10. 

17.  If  the  radii  of  two  circles  are  as  1  :  4,  and  the  area  of 
the  first  circle  is  1  square  inch,  what  is  the  area  of  the  other 
circle  ? 

18.  Given  a  circle  ;  construct  a  concentric  circle  \  as  large. 


RATIOS.  153 

Lesson  82.    Review. 

1.  Review  the  italicized  exercises  in  Lessons  78-80. 

2.  Prove  that  two  isosceles  triangles  are  similar  if  an 
angle  of  one  is  equal  to  the  homologous  angle  of  the  other. 

3.  Are  two  equilateral  triangles  similar?    Why? 

4.  The  sides  of  a  triangle  are  5,  6,  7.     Find  the  sides  of 
a  similar  triangle  if  its  ratio  of  similitude  to  the  given  tri- 
angle is  10  :  7. 

5.  Draw   an   isosceles   triangle,    and   then   construct   a 
similar  triangle  whose  ratio  of  similitude  to  the  first  triangle 
is  3  :  1 .     What  is  the  ratio  of  their  areas  ? 

6.  Compare  the  areas  of  the  two   parts   into  which  a 
parallelogram  is  divided  by  the  line  joining  the  middle  points 
of  two  adjacent  sides.     (See  p.  84,  No.  3;  p.  143,  No.  14.) 

7.  The  ratio  of  similitude  of  two  similar  triangles  is  3  :  5. 
The  first  triangle  has  an  area  of  27  sq.  ft.    What  is  the  area 
of  the  other  triangle  ? 

8.  A  vertical  rod  6  ft.  high  throws  a  shadow  4  ft.  long 
at  the  same  time  that  a  church  spire  casts  a  shadow  120  ft. 
long.     How  high  is  the  spire  ? 

9.  In  order  to  find  the  distance  from  my  position  A  to 
the  enemy's  fort  B,  I  run  a  line  AC  J_  to  AB  and  equal  to 
2000  ft.     Through  a  point  D  in  AC  100  ft.  from  C  I  run  a 
line  DE  II  to  AB,  meeting  BC  at  E.     If  DE  is  found  to  be 
800  ft.,  how  far  am  I  from  the  fort? 

10.  The  radius  of  a  circle  =  17  ft.     Through  a  point  upon 
a  diameter  9  ft.  from  the  circumference  a  chord  _L  to  this 
diameter  is  drawn.     Find  the  length  of  this  chord. 

11.  How  many  miles  is  the  light  of  a  lighthouse  200  ft. 
high  visible  at  sea?     (Radius  of  earth  =  4000  miles.) 


154  LESSONS   IN   GEOMETRY. 

Lesson  83.    Review. 

1.  Review  all  the  italicized  exercises  in  this  chapter. 

2.  What  is  the  ratio  of  4m  to  lkm  ? 

3.  What  is  the  ratio  of  5cm  to  5m? 

4.  What  is  the  ratio  of  8cm  to  24dm? 

5.  What  length  is  to  60m  as  5  is  to  6  ? 

6.  The  areas  of  two  triangles  are  480qm  and  300qm.     If 
their  bases  are  equal,  compare  their  altitudes. 

7.  If  the  altitude  of  the  larger  triangle  in  No.  6  is  30m, 
find  its  base  and  the  altitude  of  the  other  triangle. 

8.  Compare  the  areas  of  two  similar  triangles  if  the  ratio 
of  similitude  is  equal  to  ^. 

9.  From  one  corner  of  a  field  in  the  shape  of  an  equi- 
lateral triangle  I  wish  to  cut  off  a  similar  triangle  equal  in 
area  to  y^  of  the  entire  triangle.     How  must  I  run  the 
division  line? 

10.  A  rectangle  36cm  by  20cm  is  so  divided  into  three  parts 
by  lines  drawn  from  one  vertex  that  the  parts  are  as  the 
numbers  2,  3,  4.     How  great  is  each  part?     How  are  the 
sides  divided  by  the  division  lines? 

11.  Two  sides  of  a  field  in  the  shape  of  an  isosceles  right 
triangle  are  each  400m  long.     It  is  desired  to  cut  from  the 
field  a  lot  containing  one  hectar  of  land  by  a  straight  fence 
starting  from  the  vertex  of  the  right  angle.     What  portion 
of  the  hypotenuse  will  be  cut  off  by  this  fence  ? 

12.  What  is  the  mean  proportional  between  49  and  121  ? 

13.  How  high  is  a  tower  if  it  casts  a  shadow  28m  long  at 
the  same  time  that  a  vertical  staff  4m  high  casts  a  shadow 
1.45ra  long? 


CHAPTER   IX. 


SOLIDS. 

Lesson  84. 

1.  Define  a  plane  (see  p.  9,  No.  9). 

A  plane  is  named  by  three  or  more  letters  written  on  its 
boundary  lines,  or  by  a  single  letter  written  upon  its  sur- 
face ;  thus  the  plane  in  Fig.  151  is  the  plane  ABCD,  or  the 
plane  ABC,  or  simply  the  plane  M. 

2.  When  is  a  straight  line  parallel  to  a  plane  ? 

A  straight  line  and  a  plane  are  parallel  when  they  will  not 
meet,  however  far  both  are  produced. 

3.  What  is  the  foot  of  a  line  ? 

If  a  straight  line  meets  a  plane,  the  point  where  it  meets 
the  plane  is  called  its  foot. 

4.  When  is  a  straight  line  perpendicular  to  a  plane  ? 

When  it  is  perpendicular  to  every  straight  line  that  can  be 
drawn  through  its  foot  in  the  plane. 

5.  Name  in  Fig.  151  lines  II  to 
the  plane  J/,  _L  to  Jf,  oblique 
or  inclined  to  M. 

6.  Are  two  lines  necessarily 
parallel    if   they  are  J_  to   the 
same  plane?     If  they  are  II  to 

the  same  plane?  FIG.  151. 

Illustrate  your  answers  by  reference  to  Fig.  151. 


156 


LESSONS    IN    GEOMETRY. 


7.  When  are  two  planes  parallel  to  each  other? 
Two  planes  are  parallel  if  they  will  not  meet,  however  far 

both  are  produced. 

Two  parallel  planes  are  everywhere  equally  distant,  and 
their  distance  apart  is  equal  to  the  length  of  a  perpendicular 
dropped  from  any  point  in  one  of  the  planes  to  the  other  plane. 

8.  What  is  the  intersection  of  two  planes  ? 
The  intersection  of  two  planes  is  a  straight  line. 

9.  Define  a  dihedral  angle. 

If  two  planes  intersect,  the  amount  of  rotation  about  the 
intersection  required  to  make  one  of  the  planes  coincide  with 
the  other  is  called  the  dihedral  angle  of  the  two  planes. 

The  two  planes  are  the  faces  of  the  angle  ;  their  intersec- 
tion is  the  edge  of  the  angle. 

Dihedral  angles  are  expressed  in  degrees,  etc.,  like  ordi- 
nary angles. 

H  G 


D 


FIG.  152. 


FIG.  153. 


10.  Define  the  plane  angle  of  a  dihedral  angle. 

The  dihedral  angle  formed  by  two  planes  ABC,  ABE 
(Fig.  152),  is  obviously  equal  to  the  angle  formed  by  two 
lines  PQ,  PR,  drawn  one  in  each  face  from  any  point  P  of 
the  edge  and  _L  to  the  edge.  The  angle  of  any  two  lines  so 
drawn  is  called  the  plane  angle  of  the  dihedral  angle. 

Thus,  the  angles  RPQ,  CBE,  DAF,  are  plane  angles  of 
the  dihedral  angle  in  Fig.  152, 


SOLIDS.  157 

11.  W*hen  are  two  planes  perpendicular  to  each 
other? 

AVlien  their  dihedral  angle  is  a  right  angle. 

12.  Name  in  Fig.   153  two  parallel  planes,  two  perpen- 
dicular planes,  two  inclined  planes.     In  the  last  two  cases 
what  are  plane  angles  of  the  dihedral  angles? 

13.  If  two  parallel  planes  are  cut  by  a  third  plane,  what 
is  true  of  their  intersections?     Illustrate  by  Fig.  153. 

14.  If  two  planes  are  each  _L  to  a  third  plane,  what  is  true 
of  their  intersection?     Illustrate  by  Fig.  153. 

15.  How  are  planes  represented  on  paper? 

The  part  taken  for  representation  is  usually  a  rectangle. 

If  the  plane  is  to  be  represented  as  coincident  with,  or 
parallel  to,  the  plane  of  the  paper,  we  simply  draw  the  rec- 
tangle in  its  true  shape.  In  all  other  cases  we  must  substi- 
tute a  parallelogram  for  the  rectangle.  The  proper  way  to 
draw  the  parallelogram  depends  on  the  position  of  the  plane 
with  respect  to  the  eye  and  the  line  of  sight.  By  a  little 
practice  any  one  can  learn  to  draw  with  tolerable  accuracy 
planes  in  various  different  positions. 

16.  Draw  two  parallel  planes  joined  by  a  line  JL  to  them. 

17.  Draw  two  perpendicular  planes. 

18.  Draw  two  planes  forming  a  dihedral  angle  of  about  45°. 

19.  Draw  three  planes  each  J_  to  the  other  two. 

20.  Draw  three  parallel  planes,  and  a  fourth  plane  inter- 
secting them. 

21.  Draw  a  horizontal  plane  and  three  vertical  planes,  all 
passing  through  a  line  J_  to  the  horizontal  plane. 


158 


LESSONS   IN   GEOMETKY. 


Lesson  85. 

1.  Define  a  solid  (see  p.  9,  No.  9). 

2.  Define  a  cube  (Fig.  154). 

A  cube  is  a  solid  bounded  by  six  equal  squares. 
The  six  squares  are  called  the  faces  of  the  cube,  and  their 
intersections  are  called  its  edges. 

3.  Name  in  Fig.  154  the  faces  and  the  edges  that  are  (1) 
II  to  the  face  ABCD ;   (2)  _L  to  the  face  ABCD. 

H  G 


FIG. 154. 


FIG. 155. 


4.  How  is  a  model  of  a  cube  made  9 

Draw  six  equal  squares  on  stiff  cardboard,  in  one  group,  as 
shown  in  Fig.  155.  Cut  out  the  entire  group,  and  then  cut 
half-way  through  the  cardboard,  along  the  interior  lines  of 
division.  Then  fold  the  squares  over  these  lines  into  the 
shape  of  a  cube,  and  fasten  the  edges  with  slips  of  thin 
paper  and  mucilage  (or  paste) . 

5.  What  is  meant  by  developing  a  surface? 

To  develop  the  surface  of  a  solid  is  to  construct  upon  a 
plane  surface  (as  a  sheet  of  paper)  the  surfaces,  plane  or 
curved,  which  form  the  boundaries  of  the  solid. 

The  figure  thus  obtained  is  called  the  development  of  the 
surface.  Fig.  155  shows  the  development  of  the  cube. 

6.  Make  a  model  of  a  cube  (edge  3  in.). 


SOLIDS. 


159 


7.  Define  the  volume  of  a  solid. 

The  volume  of  a  solid  is  the  amount  of  space  it  fills. 

Volumes,  like  surfaces  and  lines,  are  measured  by  finding 
how  many  times  a  unit  of  volume  is  contained  in  the  solid  to 
be  measured. 

The  common  units  of  volume  are  the  cubic  inch,  the 
cubic  foot,  and  the  cubic  yard.  Each  of  these  is  a  cube 
whose  edge  is  equal  to  the  corresponding  unit  of  length. 
1  cub.  ft.  =1728  cub.  in.  ;  1  cub.  yd.  =  27  cub.  ft. 

8.  How  is  the  volume  of  a  cube  found  ? 

Suppose,  for  example,  that  one  edge  =10  in.  (Fig.  156). 
The   lower  face  ABC  may  be  di- 
vided into  10x10,  or  100,  square 
inches  (p.  000,  No.  0). 

Upon  each  square  inch  a  pile  of 
10  cubic  inches  may  be  formed  (like 
those  seen  along  the  edge  DC} . 

Hence  the  entire  cube  contains 
10  x  10  x  10,  or  1000,  cubic  inches. 

A  similar  result  is  obtained,  what-   A 
ever  value  the  edge  may  have. 

Finding  the  product  of  a  number,  when  used  three  times 
as  a  factor,  is  called  cubing  the  number ;  hence  the  result 
may  be  expressed  by  a  formula,  thus : 

Volume  of  a  cube  =  cube  of  one  edge. 
Find  the  volume  and  the  entire  surface,  having  given : 
9.    Edge  8  in.  11.    Edge  1  ft.  2  in. 

10.    Edge  9  in.  12.    Edge  2  ft.  3  in. 

13.  Find  one  edge  if  the  volume  is  8000  cubic  feet. 

14.  How  many  square  feet  of  lead  are  needed  to  line  the 
bottom  and  sides  of  a  cubical  tank  44  feet  deep,  and  how 
many  cubic  feet  of  water  will  the  tank  hold? 


FIG. 156. 


160 


LESSONS   IN    GEOMETKY. 


Lesson  86. 

1.  Define  a  rectangular  solid. 

A  rectangular  solid  is  a  solid  bounded  by  six  rectangles. 
The  rectangles  are  called  its  faces,  and  their  intersections  are 
called  its  edges. 

The  upper  and  lower  faces  are  also  called  the  bases,  and 
the  distance  between  them  the  height. 

2.  In  the  rectangular  solid  shown  in  Fig.  157,  name  the 
edges  equal  to  AB ;  to  BC ;  to  BF.  Also  name  the  faces 
and  edges  II  to  AB ;  1_  to  An. 

Q 
H G 

—  -'" ^" ^B 

/       i  T,   /i nun 

E 


FIG.  157. 

3.  Name  the  three  dimensions  of  the  solid  in  Fig.  157. 

4.  Name  in  Fig.  157  three  pairs  of  equal  and  parallel  faces. 

5.  Draw  two  rectangular  solids  differing  in  shape. 

6.  Make  a  model  of  a  rectangular  solid  6  in.  long,  4  in. 
wide,  and  3  in.  high  (see  p.  158,  No.  4). 

7.  How  is  the  volume  of  a  rectangular  solid  found  ? 
Suppose  that  the  dimensions  are  :    length,  5  in.  ;  breadth, 

3  in.  ;  height,  7  in.  (see  Fig.  158).  The  base  may  be  divided 
into  3x5,  or  15,  square  inches.  Upon  each  square  inch  there 
may  be  formed  a  pile  of  7  cubic  inches  ;  hence  the  solid  must 
contain  3  x  5  x  7,  or  105,  cubic  inches.  In  general  : 

Volume  of  a  rectangular  solid  =  length  x  breadth  x  height. 


SOLIDS.  161 

Find  the  entire  surface  and  the  volume  of  a  rectangular 
solid,  having  given  : 

8.  Length  9  in.,  breadth  7  in.,  thickness  3  in. 

9.  Length  6f  ft.,  breadth  5J  ft.,  depth  4J  ft. 

10.  Length  4£  ft.,  breadth  1|  yds.,  height  108  in. 

11.  Length  4  ft.  8  in.,  breadth  3  ft.  10  in.,  height  3  ft. 

12.  How  many  tons  of  coal  will  a  bin  20  ft.  by  16  ft.  by 
8  ft.  hold,  allowing  40  cubic  feet  to  a  ton? 

13.  A  cellar  which  measures  12  ft.  by  6  ft.  is  flooded  to  a 
depth  of  4  in.     Find  the  weight  of  the  water,  supposing  that 
1  cub.  ft.  of  water  weighs  1000  oz.  (62£  Ibs.). 

14.  What  weight  of  water  will  a  rectangular  cistern  hold, 
its  length  being  4  ft.,  breadth  2  ft.  6  in.,  depth  3  ft.  3  in., 
and  1  cub.  ft.  of  water  weighing  1000  oz.? 

15.  How  many  bricks  9  in.  by  4^  in.  by  3  in.  are  needed 
to  build  a  wall  90  ft.  long,  18  in.  thick,  and  8  ft.  high? 

16.  A  book  is  8  in.  long,  6  in.  wide,  and  l\  in.  thick. 
Find  the  depth  of  a  box  3  ft.  4  in.  long,  and  2  ft.  6  in.  wide, 
that  it  may  hold  400  such  books. 

17.  Marble  is  2.716  times  as  heavy  as  water,  and  1  cub.  ft. 
of  water  weighs  1000  oz.     Find  the  weight  of  a  block  of 
marble  9  ft.  6  in.  long,  2  ft.  3  in.  wide,  and  2  ft.  thick. 

18.  Iron  weighs  7.2  times  as  much  as  water,  and  1  cub.  ft. 
of  water  weighs  1000  oz.     What  will  an  open  cistern  made 
of  iron  1  in.  thick  weigh  when  empty,  if  its  external  dimen- 
sions are  5  ft.,  4  ft.,  and  3  ft.  ? 

19.  A  cistern  is  5  ft.  6  in.  long,  3  ft.  9  in.  wide,  and  1  ft. 
3  in.  deep.     How  many  gallons  of  water  will  it  hold  ?    What 
weight  of  water  will  it  hold  ?    (1  gal.  of  water  =  231  cub.  in.  ; 
1  cub.  ft.  of  water  weighs  1000  oz.) 


162 


LESSONS   IN   GEOMETRY. 


Lesson  87. 

1.  Define  a  right  prism,  and  terms  related  to  it. 

A  right  prism  is  a  solid  bounded  by  two  equal  and  parallel 
polygons  called  bases,  and  by  three  or  more  rectangles  called 
the  lateral  faces.  The  intersections  of  the  lateral  faces  are 
called  the  lateral  edges. 

The  distance  between  the  bases  is  called  the  height  of  the 
prism. 

A  prism  is  called  triangular  if  the  bases  are  triangles, 
hexagonal  if  they  are  hexagons,  etc. 

A  regular  prism  is  a  right  prism  whose  bases  are  regular 
polygons. 

The  prism  in  Fig.  159  is  a  regular  hexagonal  prism. 

2.  What  positions  with  respect  to  the  bases  do  the  lateral 
faces  of  a  right  prism  have?  the  lateral  edges? 

3.  Name  on  a  right  prism  the  edges  that  are  parallel. 

4.  Draw  a  triangular  prism. 

5.  Draw  a  hexagonal  prism. 


FIG.  159. 


FIG.  160. 


6.  Make  a  model  of  a  regular  triangular  prism  (Fig.  1GO). 

7.  How  is  the  volume  of  a  right  prism  found  ? 

It  can  be  proved  that  the  volume  of  a  right  prism  is  found 
by  multiplying  the  area  of  the  base  by  the  height. 

Volume  of  a  right  prism  =  base  x  JieiyJit. 


SOLIDS.  163 

NOTE.  —  The  word  "prism"  used  alone  means  here  "right  prism." 

Find   the   entire   surface    and   the  volume   of   a  prism, 
having  given : 

8.  Area  of  square  base  49  sq.  in.,  height  8  in. 

9.  Square  base,  side  of  base  4  in.,  height  10  in. 

10.  Rectangular  base  6  ft.  by  3£  ft.,  height  12  ft. 

11.  Base  an  equilateral  A,  side  2  ft.,  height  7  ft. 

12.  Base  a  regular  hexagon,  side  4  ft.  9  in.,  height  12-J  ft. 

13.  Base  a  regular  hexagon,  side  10  in.,  height  10  ft. 

14.  Is  a  cube  a  right  prism?     Why?     Is  a  rectangular 
solid  a  right  prism?     Why? 

15.  What  is  the  volume  of  a  regular  four-sided  prism  if 
the  height  is  6  in.  and  one  side  of  the  base  is  2  in.  ?     What 
would  be  the  volume  if  the  height  were  doubled?  if  the  side 
of  the  base  were  doubled  ?  if  both  were  doubled  ? 

16.  The  bases  of  a  prism  are  trapezoids  whose  parallel 
sides  are  12  ft.  and  8  ft.,  and  the  altitude  is  6  ft.     Find  the 
volume  of  the  prism  if  the  height  is  32  ft. 

17.  How  many  cubic  feet  of  stone  are  required  to  build  a 
dam  1000  ft.  long,  20  ft.  high,  10  ft.  wide  at  the  bottom,  and 
4  ft.  wide  at  the  top?    (Consider  the  dam  to  be  a  trapezoidal 
prism  like  that  in  No.  16.) 

18.  The  distance  around  a  reservoir  in  the  shape  of  a 
regular  hexagon  is  360  ft.     If  the  average  daily  loss  from 
evaporation  amounts  to  a  layer  of  water  2   in.   deep,   how 
many  cubic  feet  of  water  must  be  supplied  daily  to  maintain 
the  water  at  a  constant  level  ? 

19.  Pencils  are  often  made  in  the  shape  of  regular  hexa- 
gonal prisms.     Find  the  volume  of  the  pencil  given  you  (cor- 
rect to  a  hundredth  of  a  cubic  inch). 


164 


LESSONS   IN  GEOMETRY. 


Lesson  88. 

1.  Define  a  right  cylinder,  and  the  related  terms. 
A  right  cylinder  is  the  solid  generated   by  a  rectangle 

revolving  about  one  of  its  sides. 

A  right  cylinder  is  also  called  a  cylinder  of  revolution. 

If  the  rectangle  ABCD  (Fig.  161)  revolve  about  the 
side  CD,  the  sides  AD,  BO,  will  describe  equal  and  parallel 
circles,  and  the  side  AB  will  describe  a  curved  surface.  A 
right  cylinder,  therefore,  is  a  solid  bounded  by  two  equal 
and  parallel  circles  and  a  curved  surface  lying  between  the 
circles. 

The  circles  are  called  the  bases  of  the  cylinder,  and  the 
curved  surface  is  called  its  lateral  surface. 

The  axis  of  a  right  cylinder  is  the  line  joining  the  centres 
of  the  bases  ;  it  is  perpendicular  to  the  bases. 

The  height  of  a  cylinder  is  the  length  of  its  axis. 

2.  What  is  the  development  of  the  lateral  surface?     Illus- 
trate by  means  of  a  sheet  of  paper. 


FIG.  161. 


FIG.  162. 


3.  Draw  two  right  cylinders  differing  in  shape. 

4.  Make  a  model  of  a  right  cylinder  (see  Fig.  162) 

5.  What  are  the  general  formulas  for  finding  the 
lateral  surface  and  the  volume? 

Lateral  surface  =  circumference  of  base  x  height. 
Volume  =  base  X  height. 


SOLIDS.  165 

NOTE.  —  The  word  "cylinder"  is  here  used  meaning  "right  cylinder." 

Find  the  lateral  surface  and  the  volume  of  a  cylinder, 
having  given  : 

6.  Radius  of  base  7  in.,  height  10  in. 

7.  Radius  of  base  1  ft.  2  in.,  height  5  ft. 

8.  Diameter  of  base  9  ft.  4  in.,  height  12  ft. 

9.  Circumference  of  base  7  ft.  4  in.,  height  10  ft. 

10.  How  large  a  cylinder  can  be  made  by  rolling  up  a 
rectangular  sheet  of  tin  88  in.  by  66  in.,  so  that  the  height 
of  the  cylinder  is  88  in.?     How  large,  if. rolled  up  so  that 
the  height  is  66  in.? 

11.  Find  the  height  of  a  cylinder  if  the  volume  is  114  cub. 
yds.  2  cub.  ft.,  and  the  radius  of  base  7  ft. 

12.  How  many  cubic  yards  of  earth  must  be  dug  out  to 
make  a  well  3  ft.  in  diameter,  and  20  ft.  deep  ? 

13.  The  diameter  of  a  well  is  4  ft.  8  in.,  its  depth  is  30  ft. 
Find  the  cost  of  digging  it  at  $3.75  per  cubic  yard. 

14.  How  many  cubic  yards  of  earth  must  be  dug  out  in 
making  a  tunnel  100  yds.  long,  whose  section  is  a  semicircle 
with  a  radius  of  10  ft.  ? 

15.  What  change  in  the  volume  of  a  cylinder  is  produced 
by  doubling  its  height  ?  by  doubling  the  diameter  of  its  base  ? 
by  doubling  both  ? 

16.  Two  cylinders  have  the  same  height,  but  the  radius  of 
the  base  of  one  cylinder  is  six  times  that  of  the  other.    Com- 
pare their  volumes  ;  compare  also  their  lateral  surfaces. 

17.  Find  the  cost  of  cementing  the  side  and  bottom  of  a 
cylindrical  tank  20  ft.  deep  and  18  ft.  in  diameter,  at  32  cents 
per  square  foot. 

18.  How  many  gallons  of  water  are  there  in  a  cylindrical 
well  7  ft.  in  diameter,  if  the  water  is  10  ft.  deep  and  there 
are  1\  gals,  in  each  cubic  foot? 


166 


LESSONS    IN   GEOMETRY, 


Lesson  89. 

1,  Define  a  right  pyramid,  and  related  terms. 

A  right  pyramid  is  a  solid  bounded  by  a  polygon  and  three 
or  more  isosceles  triangles  which  have  a  common  vertex. 
The  polygon  is  called  the  base,  the  triangles  the  lateral  faces, 
and  their  common  vertex  the  vertex  of  the  pyramid. 

The  intersections  of  the  lateral  faces  are  called  the  lateral 


The  distance  from  the  vertex  to  the  base  is  called  the 
height  of  the  pyramid. 

A  regular  pyramid  is  a  right  pyramid  whose  base  is  a 
regular  polygon.  The  lateral  faces  are  equal  isosceles  tri- 
angles, and  their  common  altitude  is  called  the  slant  height 
of  the  pyramid.  Thus,  AB  is  the  slant  height  of  the  regular 
pyramid  represented  in  Fig.  163. 

2.  Draw  a  triangular  pyramid  (a  pyramid  with  a  A  for 
the  base). 

3.  Draw  a  hexagonal  pyramid. 


FIG.  163. 


FIG.  164, 


4.  Make  a  model  of  a  square  pyramid  (Fig.  164). 

5.  How  is  the  volume  of  a  pyramid  found  ? 

It  can  be  proved  that  the  volume  of  any  pyramid  may  be 
found  by  means  of  the  formula : 

Volume  of  a  pyramid  =  \  X  base  x  height. 


SOLIDS.  167 

NOTE.  —  The  word  "pyramid,"  used  alone,  means  "right  pyramid." 

6.  Find  the  volume  of  a  pyramid,  having  given  :  area  of 
the  base  =  64  sq.  in.,  height  of  pyramid  =  15  in. 

7.  Find  the  volume  of  a  pyramid  if  the  height  is  30  ft., 
and  the  base  is  a  regular  hexagon  whose  side  is  6  ft. 

8.  Find  the  slant  height  of  the  pyramid  in  No.  7. 

9.  Find  the  total  surface  of  the  pyramid  in  No.  7. 
Find  the  volume,  and  also  the  total  surface  of  a  square 

pyramid,  having  given : 

10.  Side  of  base  40  ft.,  height  48  ft. 

11.  Side  of  base  22  ft.,  height  60  ft. 

NOTE.  —  The  frustum  of  a  pyramid  is  the  portion  of  the  pyramid 
contained  between  the  base  and  a  plane  parallel  to  the  base. 

The  base  and  the  section  made  by  the  plane  are  called          ^ 
the  bases  of  the  frustum.  II  !\\ 

The  distance  between  the  bases  is  the  height. 

To  find  the  volume  of  a  frustum,  add  together  the  areas  of 
the  bases  and  the  square  root  of  their  product,  and  multiply  the 
sum  by  |  the  height.  FlG- 165- 

12.  What  kind  of  figures  are  the  lateral  faces  of  a  frustum  ? 

13.  Find  the  total  surface  of  a  frustum  of  a  square  pyr- 
amid (Fig.  165)  if  the  sides  of  the  bases  are  12  in.  and  4  in., 
and  the  slant  height  (or  altitude  of  each  face)  is  5  in. 

14.  Find  the  volume  of  the  frustum  in  No.  13  if  the  height 
of  the  frustum  is  3  in. 

15.  Find  the  volume  of  the  frustum  of  a  square  pyramid 
if  the  sides  of  the  bases  are  21  yds.  and  15  yds.,  and  height 
84  yds. 

16.  A  church  spire  has  the  shape  of  a  frustum  of  a  regular 
hexagonal  pyramid;  each  side  of  the  base  is  5  ft.,  and  of 
the  top  2  ft.  ;  the  altitude  of  each  trapezoidal  face  is  20  ft. 
How  many  square  feet  of  tin  roofing  are  required  to  cover 
the  lateral  faces  and  the  top  ? 


168 


LESSONS  IN   GEOMETRY. 


Lesson  9O. 

1.  Define  the  right  cone,  and  related  terms. 

A  right  cone  is  the  solid  generated  by  a  right  triangle 
revolving  about  one  of  its  legs. 

A  right  cone  is  also  termed  a  cone  of  revolution. 

If  the  right  triangle  ABC  (Fig.  166)  revolve  about  the  leg 
BC,  the  other  leg  AB  will  describe  a  circle  called  the  base 
of  the  cone,  and  the  hypotenuse  AC  will  describe  a  curved 
surface  called  the  lateral  or  convex  surface  of  the  cone. 

The  point  C  is  called  the  vertex  of  the  cone. 

The  leg  jBC,  about  which  the  triangle  revolves,  is  called 
the  axis  of  the  cone. 

The  length  of  the  axis  is  called  the  height  of  the  cone. 

The  length  of  the  hypotenuse  AC  of  the  generating  tri- 
angle is  called  the  slant  height  of  the  cone. 

2.  Draw  two  right  cones  differing  in  shape. 


FIG.  166. 


FIG.  167. 


3.  What  kind  of  a  figure  is  the  development  of  the  lateral 
surface  of  a  right  cone  (see  Fig.  167)? 

4.  Make  a  model  of  a  right  cone. 

5.  What  are  the  formulas  for  finding  the  lateral 
surface  and  the  volume? 

Lateral  surface  =  £  X  circumference  of  base  x  slant  height. 
Volume  =  1  X  base  x  height. 


SOLIDS.  169 

NOTE.  —  The  word  "cone,"  if  used  alone,  means  a  "right  cone." 

6.  The  height  of  a  cone  is  6  in.,  the  radius  of  the  base 
is  2£  in.     Find  the  slant  height  (p.  126,  No.  1). 

7.  Find  the  volume  and  the  lateral  surface  of  a  cone  if 
the  height  is  40  ft.,  and  the  radius  of  the  base  is  9  ft. 

8.  Find  the  volume  and  the  lateral  surface  if  the  height 
is  28  ft.,  and  the  radius  of  the  base  is  21  ft. 

9.  A  right  triangle  whose  legs  are  3  in.  and  4  in.  gene- 
rates a  cone  by  revolving  about  its  shorter  leg.     Find  the 
volume  and  the  lateral  surface  of  this  cone. 

10.  Solve  No.  9  if  the  triangle  revolve  about  the  other  leg. 

11.  How  much  canvas  is  required  to  make  a  conical  tent 
80  ft.  high,  and  70  ft.  in  diameter  at  the  base? 

NOTE.  —  The  frustum  of  a  cone  is  the  part  contained 
between  the  base  and  a  plane  II  to  the  base. 

The  bases  and  height  are  defined,  and  the  volume 
found,  as  in  the  case  of  the  frustum  of  a  pyramid  (p.  167). 

The  lateral  surface =  half  the  product  of  the  circumference 
...  j    i      ;     .  »    •  T.  '  FIG.  168. 

of  base  and  the  slant  height. 

12.  How  many  square  feet  of  tin  will  be  required  to  make 
a  funnel  with  the  radii  of  top  and  bottom  14  in.  and  7  in. 
respectively,  and  the  height  24  in.  ? 

13.  Find  the  expense  of  polishing  the  curved  surface  of  a 
marble  column  in  the  shape  of  a  frustum  of  a  right  cone, 
slant  height  12  ft.,  radii  of  bases  3  ft.  6  in.  and  2  ft.  4  in., 
at  60  cents  per  square  foot. 

14.  A  round  stick  of  timber  is  20  ft.  long,  3  ft.  in  diameter 
at  one  end,  2.6  ft.  at  the  other.     How  many  cubic  feet  does 
it  contain  ? 

15.  A  bucket  is  16  in.  deep,  18  in.  wide  at  the  top,  and 
12  in.  wide  at  the  bottom.     How  many  gallons  of  water  will 
it  hold,  reckoning  7£  gallons  to  the  cubic  foot? 


170  LESSOMS   IN   GEOMETRY. 

Lesson  91. 

1.  Define  "a  sphere. 

A  sphere  is  the  solid  generated  by  a  semicircle  revolving 
about  its  diameter. 

If  the  semicircle  ACBO  (Fig.  169)  revolve  about  the 
diameter  AB,  the  semicircle  generates  a  sphere,  and  the 
semi-circumference  ACB  generates  a  curved  surface  forming 
the  boundary  of  the  sphere.  The  centre  0  of  the  semicircle 
is  the  centre  of  the  sphere.  All  points  in  a  spherical  surface 
are  equidistant  from  the  centre. 

2.  Define  a  radius  and  a  diameter  of  a  sphere. 
In  Fig.  169  OA  is  a  radius,  and  AOB  is  a  diameter. 

A 


3.  Define  great  circles  and  small  circles. 

Every  section  of  a  sphere  made  by  a  plane  is  a  circle.  If 
the  plane  pass  through  the  centre,  the  circle  is  called  a  great 
circle ;  in  all  other  cases  the  circle  is  called  a  small  circle. 

4.  Define  the  axis  and  poles  of  a  circle. 

The  axis  of  a  circle  (great  or  small)  is  the  diameter  which 
is  perpendicular  to  the  plane  of  the  circle. 
The  poles  of  a  circle  are  the  ends  of  its  axis. 

5.  Illustrate  the  nbovc  definitions  by  referring  to  Fig.  1  70. 


SOLIDS.  171 

6.  State  three  truths  respecting  great  eircles. 

(1)  A  great  circle  divides  a  sphere  into  two  equal  parts. 
These  two  equal  parts  are  called  hemispheres. 

(2)  Two  great  circles  bisect  each  other. 

(3)  The  poles  of  a  great  circle  are  at  the  distance  of  a 
quadrant  from  every  point  in  its  circumference. 

7.  Define  parallels. 

If  a  sphere  is  cut  by  a  series  of  parallel  planes,  the  cir- 
cumferences of  the  circles  thus  formed  are  called  parallels. 
Example :  parallels  of  latitude  on  the  earth's  surface. 

8.  Define  a  zone  and  its  altitude. 

A  zone  is  the  portion  of  the  surface  of  a  sphere  contained 
between  two  parallel  planes.  The  distance  between  the  two 
planes  is  the  altitude  of  the  zone. 

Example:  the  torrid  zone,  etc.,  on  the  earth's  surface. 

9.  Define  meridians* 

If  a  sphere  is  cut  by  a  series  of  planes  all  passing  through 
the  same  diameter,  the  circumferences  of  the  great  circles 
thus  formed  are  called  meridians. 

Example :  meridians  of  longitude  on  the  earth's  surface. 

10.  Define  a  lune. 

A  lune  is  the  portion  of  the  surface  of  a  sphere  contained 
between  two  semi-circumferences  of  great  circles. 

11.  Point  out  in  Fig.  170  parallels,  meridians,  a  zone,  its 
altitude,  a  lune. 

12.  Draw  Fig.  170  (to  a  larger  scale  if  you  prefer). 

13.  Draw  a  sphere  cut  by  three  planes  passing  through 
the  centre,  and  each  perpendicular  to  the  other  two. 

14.  If  (Fig.  170)  PEQ  represent  the  meridian  through 
Greenwich,    point   out   the    latitude   and    longitude   of   the 
point  M. 


172 


LESSONS   IN   GEOMETRY. 


Lesson   92. 

1.  When  is  a  sphere  inscribed  in  a  cylinder? 

A  sphere  is  inscribed  in  a  cylinder  when  its  surface  touches 
the  bases  and  the  lateral  surface  of  the  cylinder.  Also,  the 
cylinder  is  circumscribed  about  the  sphere  (Fig.  171). 

2.  Compare  the  radius  and  diameter  of  a  sphere  with  the 
dimensions  of  the  circumscribed  cylinder. 

P 


FIG.  171. 


3.  How  is  the  surface  of  a  sphere  found  ? 

It  can  be  proved  that  the  surface  of  a  sphere  is  equal  to 
the  lateral  surface  of  the  circumscribed  cylinder. 

Let  r  denote  the  radius  of  a  sphere.  Then  r  is  also  equal 
to  the  radius  of  the  base  of  the  circumscribed  cylinder,  and 
2r  is  its  height;  therefore  its  lateral  surface  is  equal  to 
2-n-r  x  2r,  or  ±irr  (p.  164,  No.  5). 

Hence  if  the  value  of  TT  be  taken  as  \2-, 

Surface  of  a  sphere  =  -8T8-  r2. 

4.  How  is  the  volume  of  a  sphere  found  ? 

The  volume  of  a  sphere  is  found  by  multiplying  the  sur- 
face by  one-third  of  the  radius.  Therefore  if  r  denote  the 
radius,  the  volume  =  \r  X  4^=  |  Trr*. 

Hence  if  the  value  of  TT  be  taken  as      , 


SOLIDS.  173 

Find  the  surface  and  the  volume  of  a  sphere,  given : 

5.  Radius  1  in.  10.    Diameter  7  in. 

6.  Radius  2  in.  11.    Diameter  16  in. 

7.  Radius  3  in.  12.    Diameter  21  in. 

8.  Radius  4  in.  13.    Diameter  42  ft. 

9.  Radius  3  ft.  6  in.  14.    Diameter  5  ft.  10  in. 

15.  A  regulation  base-ball  is  9J  in.  in  circumference.    How 
many  square  inches  of  leather  are  required  in  order  to  make 
1000  base-balls? 

16.  The  circumference  of  a  dome  in  the  shape  of  a  hemi- 
sphere is  66  ft.    How  many  square  feet  of  tin  roofing  are 
required  to  cover  it? 

17.  If  the  ball  on  the   top   of  St.  Paul's  Cathedral   in 
London  is  6  ft.  in  diameter,  what  would  it  cost  to  gild  it  at 
7  cents  per  square  inch? 

18.  The  area  of  a  zone  is  equal  to  the  product  of  its  alti- 
tude and  the  circumference  of  a  great  circle. 

Find  the  area  of  the  upper  zone  on  the  sphere  in  Fig.  172 
if  the  altitude  PR  is  15  ft.  and  the  radius  is  35  ft. 

19.  The  altitude  of  the  torrid  zone  on  the  earth  is  about 
3200  miles.     What  is  its  area  in  square  miles,  assuming  the 
radius  of  the  earth  to  be  equal  to  4000  miles  ? 

20.  If  one  cubic  inch  of  iron  weighs  4J  oz.,  what  will  an 
iron  ball  10J  in.  in  diameter  weigh? 

21.  Find  the  weight  of  a  spherical  shell  10  in.  in  diameter 
and  2  in.  thick,  composed  of  a  substance  1  cub.  ft.  of  which 
weighs  216  Ibs. 

22.  How  much  rubber  is  there  in  a  tennis  ball  whose 
diameter  is  3^  in.  if  the  thickness  of  the  rubber  is  J  in.  ? 

How  much  cloth  is  needed  to  cover  the  ball? 
How  much  pasteboard  is  needed  to  make  a  cylindrical  box, 
open  at  the  top,  which  will  just  hold  the  ball? 


174  LESSONS  IN  GEOMETEY. 

Lesson  93.    Review. 

NOTE.  —  1  cub.  ft.  of  water  weighs  1000  oz. ;  16  oz.  =  1  Ib. 

1.  Review  the  italicized  exercises  in  Lessons  84-88. 

2.  Draw  two  parallel  planes  cut  by  a  third  plane.    What 
is  true  of  the  two  lines  of  intersection  ? 

3.  Draw  two  planes  each  perpendicular  to  a  third  plane. 
What  is  true  of  their  intersection  ? 

4.  Find  the  volume  of  the  cube  given  you. 

5.  Find  the  volume  of  the  rectangular  solid  given  you. 

6.  Find  the  volume  of  the  prism  given  you. 

7.  Find  the  volume  of  the  cylinder  given  you. 

8.  What  will  be  the  cost  of  digging  a  cellar  36  ft.  long, 
20  ft.  wide,  and  8  ft.  deep,  at  5  cents  per  cubic  foot? 

9.  What  will  it  cost  to  sink  a  well  100  ft.  deep  and  3£  ft. 
in  diameter,  at  $2  per  cubic  foot? 

10.  If  a  cubic  foot  of  brass  is  drawn  into  a  wire  -^y  of  an 
inch  in  diameter,  what  will  be  the  length  of  the  wire? 

11.  How  many  cylindrical  pieces  of  lead  f  of  an  inch  in 
diameter  and  ^  of  an  inch  thick  must  be  melted,  in  order  to 
form  a  cube  whose  edge  is  3  in.  long? 

12.  The  piston  of  a  pump  is  14  in.  in  diameter,  and  moves 
through  a  space  of  3  ft.     How  many  tons  of  water  will  be 
thrown  out  by  1000  strokes? 

13.  A  body  is  placed  under  water  in  a  right  cylinder 
60  in.  in  diameter,  and  the  level  of  water  is  observed  to 
rise  30  in.     Find  the  volume  of  the  body. 

14.  How  much  will  a  brass  cylinder  weigh  under  water,  if 
the  height  is  10  in.  and  the  diameter  of  the  base  is  7  in.  ? 
Brass  is  7.8  times  heavier  than  water,   and  a  body  when 
immersed  in  water  loses  a  weight  equal  to  the  weight  of  the 
water  displaced. 


SOLIDS.  175 

Lesson  94.     Review. 

1.  Review  the  italicized  exercises  in  Lessons  89-92. 

2.  Find  the  volume  of  the  pyramid  given  you. 

3.  Find  the  volume  of  the  cone  given  you. 

4.  Find  the  volume  of  the  sphere  given  you. 

5.  Make  a  model  of  a  regular  pyramid  bounded  by  four 
equal  equilateral    triangles.       Make 

each  edge  4  in.  long. 

Then   find    the   volume    and    the       .\ ^         ///// 

/  \  /  \  m/t 

entire  surface. 

(The  apothem  of  the  base  may  be 

FIG.  173. 
found  from  the  table  on  p.  122.) 

6.  If  the  height  of  a  cone  and  the  radius  of  its  base  are 
known,  how  is  the  slant  height  found? 

7.  Can  the  surface  of  a  sphere  be  developed? 

8.  How  much  asphalt  varnish  will   be  needed  to  coat 
1000  spherical  bombs,  each  10£  in.  in  diameter,  if  1  pint  of 
varnish  will  cover  300  square  inches  of  surface  ? 

9.  If  a  cylinder  and  a  cone  have  equal  bases  and  equal 
volumes,  compare  their  heights. 

10.  Find  the  volume  of  a  cylinder  circumscribed  about  a 
sphere  whose  radius  is  7  in. 

11.  A  cylindrical  pail  is  partly  filled  with  water.     The 
radius  of  its  base  is  3^-  in.     If  a  bullet  3^  in.  in  diameter  be 
dropped  in,  how  much  will  the  level  of  the  water  rise? 

12.  The  chimney  of  a  factory  has  the  shape  of  the  frustum 
of  a  square  pyramid;  its  height  is  180  ft.,  and  the  sides  of 
its  upper  and  lower  bases  are  16  ft.  and  10  ft.  respectively  ; 
the  section  of  the  flue  is  throughout  the  entire  length  a  square 
whose  side  is  7  ft.     How  many  cubic  feet  of  brick  does  the 
chimney  contain? 


176  LESSONS   IN  GEOMETRY. 

Lesson  95.    Review. 

1.  What  are  the  chief  metric  units  of  volume  ? 

The  cubic  meter,  having  the  abbreviation  cbm. 

The  cubic  decimeter,  having  the  abbreviation  cdm. 

The  cubic  centimeter,  having  the  abbreviation  com. 

The  cubic  meter,  when  used  for  measuring  wood,  is  called 
a  stere ;  and  the  cubic  decimeter,  when  used  for  measuring 
fluid  substances,  is  called  a  liter. 

2.  How  are  the  metric  units  of  volume  related  ? 
Since  10  centimeters  =1  decimeter,  it  follows  that 

10  x  10  x  10,  or  1000,  cubic  centimeters  =  1  cubic  decimeter. 

And  since  10  decimeters  =  1  meter,  it  follows  that 
10x10x10,  or  1000,  cubic  decimeters  =1  cubic  meter. 

The  reason  is  evident  from  a  study  of  Fig.  156,  p.  159. 

NOTE.  —  lcbm  =  35i  cub.  ft.,  nearly;  1  liter  =1^  qts.,  nearly. 

3.  How  many  cubic  centimeters  make  1  cubic  meter? 

4.  Define  the  gram  and  the  kilogram. 

A  gram  is  the  weight  of  lccm  of  pure  cold  water. 
A  kilogram  is  1000  grams,  therefore  equal  to  the  weight 
of  1000ccm,  or  1  liter,  of  pure  cold  water. 

NOTE.  —  29  grams  =1  oz.,  nearly;  1  kilogram  —  2£  Ibs.,  nearly. 

5.  What  is  the  weight  of  lcbm  of  water? 

6.  A  bottle  has  a  capacity  of  half  a  liter.     How  many 
grams  of  water  will  it  contain? 

7.  The  volume  of  a  vessel  is  2.4cbm.     What  is  the  volume 
in  liters?     What  weight  of  water  will  it  hold? 

8.  A  gallon  jug  contains  231  cub.  in.     Taking  1  inch  :is 
equal  to  \  of  a  centimeter,  find  what  weight   of  water  in 
gnuns  can  be  put  into  the  jug. 


SOLIDS.  177 

NOTE.  —  In  the  following  exercises  take  TT  equal  to  3.1416. 

9.    Find  the  total  surface  of  a  square  pyramid  if  a  side  of 
the  base  is  3m  and  the  slant  height  is  15m. 

10.  Find  the  total  surface  of  a  cone  if  its  height  is  12m  and 
the  radius  of  the  base  5m. 

11.  If  the  radius  of  a  sphere  is  8m,  find  the  area  of  a  zone 
whose  height  is  equal  to  half  the  radius. 

12.  How  high  must  a  box  5dm  long  and  2dm  wide  be  in 
order  that  it  may  hold  exactly  30  liters  ? 

13.  A  cylindrical  pail  holding  just  1  liter  is  18cm  high. 
What  is  the  diameter  of  its  base? 

14.  A  vessel  has  the  shape  of  a  frustum  of  a  cone.     The 
height  is  lm,  and  the  inside  circumferences  of  the  bases  are 
5m  and  4m.     How  many  kilograms  of  water  will  the  vessel 
hold? 

15.  What  is  the  volume  of  the  largest  sphere  that  can  be 
turned  from  a  wooden  cube  whose  edge  is  ldm? 

16.  What  is  the  volume  of  the  largest  sphere  that  can  be 
turned  from  a  wooden  cylinder,  if  the  height  of  the  cylinder 
and  diameter  of  the  base  are  each  equal  to  12cm? 

17.  Marble  is  2.7  times  as  heavy  as  water.     What  is  the 
weight  of  a  block  of  marble  20dm  by  16dm  by  8dm? 

18.  An  immersed  body  is  buoyed  up  by  the  weight  of  its 
own  volume  of  water.     Iron  is  1\  times  heavier  than  water. 
What  will  an  iron  ball  whose  diameter  is  10cm  weigh  under 
water  ? 

19.  Eighty  bullets,  equal  in  size,  are  dropped  into  a  cylin- 
drical vessel  68cm  in  diameter  containing  water.     The  level 
of  the  water  rises  2cm.     Find  the  diameter  of  a  bullet. 

20.  Lead  is  about  11 J  times  heavier  than  water.    Find  the 
weight  of  a  lead  pyramid  if  the  base  is  a  square  whose  side 
is  40cm,  and  the  lateral  faces  are  equilateral  triangles. 


178  LESSONS   IN   GEOMETRY. 

Lesson  96.     Review. 

NOTE.  —  When  ?r  is  used,  take  IT  equal  to  3.1416. 

1.  How  many  square  meters  of  sheet  iron  are  needed  to 
make  a  cylindrical  pipe  12m  and  40cm  in  diameter? 

2.  How  many  liters  of  water  are  there  in  a  cubical  tank  if 
one  edge  is  24m,  and  the  water  is  4m  deep? 

3.  A  square  field  contains  4  hectares.     What  will  it  cost 
to  dig  all  around  the  field  a  ditch  2m  deep  and  lm  wide,  the 
terms  being  25  cents  per  cubic  meter  of  earth  thrown  out? 

4.  A  Dutch  windmill,  in  the  shape  of  the  frustum  of  a 
cone,  is  12™  high.     The  outer  diameters  of  the  bases  are  16m 
and  12m;    the  inner  diameters,  12m  and   10m.     How  many 
cubic  meters  of  stone  were  needed  to  build  it? 

5.  The  piston  of  a  pump  is  36cm  in  diameter,  and  moves 
through  a  space  50cm.     How  many  kilograms  of  water  are 
thrown  out  by  1000  strokes? 

6.  A  body  is  placed  under  water  in  a  cylinder  60cm  in 
diameter.     The  level  of  the  water  is  observed  to  rise  30cm. 
Find  the  volume  of  the  body. 

7.  How  much  will  a  brass  cylinder  weigh  under  water  if 
its  height  is  64cm,  and  the  diameter  of  its  base  40cm  ? 

Brass  is  8.4  times  heavier  than  water.    (See  p.  177,  No.  20.) 

8.  When  a  body  floats  in  water,  the  weight  of  the  water 
displaced  is  just  equal  to  the  weight  of  the  body. 

Find  the  weight  of  a  sphere  20cm  in  diameter,  which  floats 
half  under  water  and  half  above. 

9.  A  cone  of  loaf-sugar  is  lm  high,  and  the  diameter  of 
its  base  is  40cm.     How  far  from  the  base  must  the  cone  be 
cut  by  a   plane  parallel  to  the  base,  in  order  that  the  two 
parts  may  be  equal  in  volume? 


VERSITY 

DRAWING  EXERCISES. 


i. 


3. 


4. 


6. 


180 


DRAWING   EXERCISES.       - 


7. 


8. 


n 


n 


9. 


DRAWING   EXEECISES. 


181 


11. 


12. 


182 


DRAWING   EXERCISES. 


13. 


f 


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